The whole idea of "being at a distance to her boundary" is misleading imho. It should correctly be (but this is much less tangible):
"being at a distance to the process of behavioural self-mediation between herself as a living body and her surroundings that takes place at her boundary"
(and thus aware of it while all this is still made possible and being carried by the physical body and its organisation)
This answer is to be understood as complementary to the answer of @GeoffreyThomas. This one delivers a good, sourced description of excentric positionality and its non-physical nature. But it cannot explain what all this "means", i.e. explain what (excentric) positionality really is and why it may be necessary to introduce it.
I'll just copy/paste parts of my thesis (which I should write on right now instead) here. References are to the German 1975 edition of his The Levels of the Organic and the Human [Die Stufen des Organischen und der Mensch] from 1928, where the concept was methodologically and materially developed. The page numbers will also be used in the upcoming English translation by Millay Hyatt.
Positionality and its meaning cannot be understood without taking into consideration how it is developed over the course of his investigation of all of nature, i.e. by merely looking at the human.
Mind, this is but an aspect of the whole of the book (and my thesis), a mere step in an elaborated argument. Even if I gave my best to render it in an understandable fashion here, it may still appear incomprehensible without considering the whole of the book/thesis.
This intricacy is due to the writing style of Plessner: as Scott Davis points out, much of the content and the extent of explanation and clarification given is dependent on the context and place within the text, so that for a full understanding of a particular point he is making, it may be necessary to read the whole text and understand the intercontextual relations (Davis, Scott (2015): 'Plessner’s Conceptual Investigations of ‘Life’: Structural Narratology', in: P. Honenberger (ed.), Naturalism and Philosophical Anthropology. London: Palgrave Macmillan UK, p 124).
Thus, this answer is very long (TL;DR at the end) and still superficial.
Especially the difference between spatial relation (in physical space) and spacelike relation (a relation that can be phenomenologically described in spatial terms but is not spatially observable) is crucial for understanding positionality.
Also, this answer necessarily is more doing philosophy than it can build on philosophy because there is no English literature as of yet (a commentary on the Levels is forthcoming) that can fully account for your question.
What is this all about? The nature of thinghood in appearance as an epistemological problem
(This is but part of his setup, but I find it to be tangible. The underlying problem is to understand both the duality described now and the historicity of the human as expressions of a single state of affairs, i.e. aspects of the very being of the human.)
The phenomenal access to something seems to be fundamentally mediated through the self (ibid:45), nothing can become a phenomenon or appear without being the object of a subject, i.e. without opposition (ibid:46). If that is true, all that appears does so to us, i.e. a phenomenon insofar it is “sensuous intuitive data [sinnlich-anschauliche Daten]” (ibid:82) necessarily has to bear perspectivity, one-sidedness.
At the same time, every phenomenon has a character of transgredience, i.e. it points towards something that lies beyond itself, beyond the particular appearance:
Firstly, appearances have a moment that implies that they are the appearance of something, i.e. that there is something it is the appearance of. All appearance seems to be in ”the state of being-carried” (ibid) by being, points towards the connectedness to a substantial core of being. Because of this dependence between the core as a carrier and the appearing properties being carried, this core itself can never appear, as it is what all appearance rests on, its “background” (ibid:87).
Secondly, as appearance of a thing, it points to the existence of other appearances of – and perspectives on – the same thing. In short: Appearances of a thing point into and around a thing (ibid:82–83).
The realness, the presence of the thing and the full unity of its image always includes “this dually directed giving of the gaze” (ibid:83).
The substantial being that is implied by this transgredience, the “core” that is the “axis” at which the appearances as phenomena of one and the same thing hinge (ibid), somehow seems to be “inside” of it, “behind” the appearances but cannot become content of phenomenal reality even by looking inside or behind the thing (ibid:85–87):
For the structure of central core content/property-carrying sides is one of fundamentally divergent spheres of objects which are, by their nature, never mutually convertible. The object is not destroyed by the schism of an interior that never appears, i.e., never becomes an outside, and of an outside that never becomes core content. (ibid:88)
Our “intuitive habitus of objects” (ibid:81) results in a schism between the divergent spheres of the outer and inner being of things, they appear in the “dual aspect” of outer and inner (ibid). This poses an epistemological problem that extends even into self-perception: As things are for us only insofar they appear to us – i.e. the world is only present to us in the form of a phenomenon, a “system of qualities” (ibid:43,45) – all that there is has to be mediated through our self. This is what Plessner calls the proposition of immanence:
The proposition of immanence means that the manifestation of any existing thing corresponds to an act of subjective attention by virtue of which it exists. (ibid:56)
This seems to imply an existential schism between subjective reality as constituted by appearances and an objective reality of the being of the thing as it “really” is (ibid:43), between appearance of things and Kantian things in themselves.
Thus, there seems to be a division of reality into “inner” (core-like, substantial, objective) and “outer” (intentional, subjective, apparent) being of a thing – of an inner and an outer layer – following from the proposition of immanence (ibid:49, 58). As they can never be mutually convertible (ibid:81–82,88), they result in a dualism that is grounded in the way we perceive the world.
But at the same time, this dualism is not apparent (sic!) in common phenomenal intuition, where we simply do see “whole things”: it is only the outcome of philosophical analysis (ibid:88–89).
What is positionality?
Positionality is a category, an idea to enable us to understand a phenomenal aspect of a thing as a real part of it, i.e. being in accord with its physical existence. As such, it is part of a hermeneutics.
The phenomenon in question is that living things, as opposed to non-living things, somehow mediate their relation to their surroundings as part of their being. This is what Plessner describes as Grundsachverhalt [underlying state of affairs] of living things, which he renders in terms of the living thing realising its own boundary, i.e. having the boundary as a property of its own being.
Positionality is a necessary character in the deduction as it alone allows a “balance” [Ausgleich] between the boundary of a living being having a) the inuitional/phenomenal aspect of realisation of its own boundary (the Grundsachverhalt), and b) its physical aspect as an observable, spatiotemporal contour (ibid:127–30).
The realisation of its boundary as a property of the living thing
What is a boundary? The boundary is that where there is a passing over between the fundamentally diverging spheres of the inner and outer (ibid:100). The boundary of a thing can have two relations to the body it encloses:
Firstly, it may not belong to the being in question. Phenomenally, the boundary then shows as the edge [Rand] of the thing that forms its gestalt or contour. In this case, it does have a phenomenal boundary, but this boundary is not an independent aspect of its being, it is the mere in-between, the virtual medium of passing over from one being to the other, spatially marked by its boundary contour. As such, it is a mere phenomenal property and only part of the outer layer of the being (ibid:101–2). The thesis says that this is the case for all non-living things.
The Grundsachverhalt instead implies that the boundary belongs to the living being, which means that the passing over from inner to outer has to become part of its "inner" being as well (ibid:104).
Positionality as a mediating category
Given that the passing over from inner to outer as directions, the function of the boundary, is carried by the living being as a property, this would have to show in the appearance of the contour and the physical boundary and representation of the outer layer as well (ibid:127–28).
The first question to be answered is: How can this appear in terms of phenomenal conditions? Plessner describes it as an actual passing over of the physical body: The body would have to show the directional crossing of its boundary to the outside and into itself. The body itself has to be “both out beyond and over against this body” (ibid:128). This going out beyond and over against would have to show at the edge, the contour, as properties can only show at the “outside” of a physical thing (ibid:83–85,128).
This leads us to the second question to be answered: How can the physical body with its observable contour appear as going out beyond and over against itself? Plessner formulates this problem as follows:
As a physical body, the thing already “is” of its own accord; being does not confront it in any way or set itself apart from it as something that exists. Because the boundary, however, belongs to that which exists, this existent becomes something that passes over in two directions. It is in this sense that it is “lifted up”—we are compelled to follow this image. It cannot remain “lifted up” though, as this would violate the determination that it remains an existing bodily thing despite “passing over.” (ibid:129)
It seems as if the conditions of the appearance of the boundary as a property of the thing and the physical determination of the thing as a thing with a definite contour are running contrary to one another. This tension is mediated by the idea of a “positional character” of the living thing: by understanding the living thing as a being that is posited [gesetzt], we gain the means to understand both spheres of being as accounted.
That positionality is a category of life can thus be understood as well: only because of positionality (or under the aspect of it), it is both possible for the physical body to bear the (purely phenomenal) dynamic Grundsachverhalt at its principally static contour as its property – per the moment of determination that is entailed in being positioned and thus having a position – and it is a necessary condition of the phenomenal appearance of the Grundsachverhalt, a structural character that allows the physical being to appear as going out beyond and over against itself, i.e. the dual aspect appears as a property of the living thing – per the moment of being lifted up. (ibid:129–30).
Positionality, core, and having a place
As seen earlier, all phenomenal properties - as properties of things - have a relation of transgredience to the core of the thing they are properties of. If the Grundsachverhalt is true, this results in the peculiar situation that the dual aspect (an outer phenomenal layer of the thing and an inner, substantial core that never itself appears) which is constituted by this transgredience is both a property of the thing and appears as its property.
While in the appearance of things the dual aspect is something that all things show as thinghood is constituted “by virtue of the dual aspect” (ibid:89), the dual aspect as a property and thus “in an essential connection with the gestalt (contour) of the body” (ibid:104) changes the nature of the transgredience of this property and its implications: the transgredience points in diverging directions; “on the one hand, it posits it out beyond it (strictly speaking: outside of it); on the other, it posits it into it (inside it)” (ibid:129).
In other words: If transgredience in a direction (towards the core) is a structural characteristic of all properties appearing as properties of a thing, the dual aspect (which is nothing more than the intuitive structure of two diverging directions) as a property results in transgredience in two opposing directions per the same structural characteristic.
Positionality thus entails two opposing directions of the spacelike relation of transgredience of the property that lies on the outside of the thing to the core. This has a twofold consequence:
Firstly, the direction “into it” leads to the core itself becoming posited by virtue of positionality (ibid:130,154–60). This happens since the dual aspect as a property of course has the transgredient character of constituting thinghood by pointing further towards a core as well. If this spacelike outer-to-inner relation becomes bidirectional and gains the character of a dual aspect, the core itself becomes “lifted up” against the properties it carries just like in the description of the boundary before.
Therefore, understanding “having the dual aspect as a property” under the aspect of positionality in this context means that the core itself, the living thing in its very being becomes posited or, which is to say the same: the living being itself is posited in opposition (i.e. as having a relation) to the properties it carries and thus becomes the subject of the life that has its body as an object (ibid:156–60,185–89).
Secondly, the direction “out beyond it” constitutes an equally spacelike relation to the surroundings. The living being “is into space”, sets its being in relation to a positional field that surrounds it and thus “has” a “natural place” that it “claims” (ibid:cf.131–32).
But these directions are only aspects of a single dual aspect, i.e. the positing into the core (into it) and into the positional field (outside beyond it) are in truth a single, bidirectional spacelike relation between the core – living being itself – and the surroundings in which it is. Thus, the bidirectional spacelike (not spatial!) relation between the outer and inner of the living being that is always “passing through” (ibid:170) the core and constitutes a field in which it is positioned and into which it positions itself (ibid:131).
This is yet another description of “going out beyond and over against itself”, but turned on the being itself and not only its boundary, and more pointedly allows for this to be understood as a state of affairs that has to be distinguished from the merely phenomenal characteristics of living beings and from the spatial relations the physical body has to its surroundings (just like non-living physical bodies) (ibid:132).
Positionality and its physical conditions: Organisation
Positionality as a mediating idea revolved around the question how the phenomenal aspect of the Grundsachverhalt – the realisation of the boundary or, which is to say the same, the dual aspect as a property – is to bring in accordance with the physical means of expression the living body has at its phenomenal boundary.
The further application of the idea of positionality on the phenomenal structure of thinghood under the aspect of the Grundsachverhalt resulted in a spacelike relatedness of the posited core to the physical body and its surroundings in which it is posited. Organisation now is “the self-mediation of the unity of the living body by its parts” (ibid:185).
Turned on what was just said about the characteristics of positionality, organisation can be understood as revolving around the question how the relatedness of the posited core to the physical body (the unity to its parts) and its surroundings (unity to positional field) that is immanent in positionality can in turn itself be understood as a) being realised by the living being physically (in relation to itself and its surroundings) and b) showing in the appearance of the physical body.
Organs as mediating whole and parts
The physical manifestation and epitome of something that is both part of and a representation of the whole is the organ (ibid). It is a mediation between the “subject” of living being – the unity (“for itself”) – and the “object” of the living being – the unity of its parts – by constituting a “third” unity, the unity of the whole being in all of its parts, i.e. its organs:
[The living thing] is doubled in itself, but unified in this doubling: unity for itself (core, the subject of having), unity in the manifold parts (functional unity, gestalt, overall function that is greater than the sum of its parts, object of having), and unity in every part (ibid:187)
The whole of the organism when (in the aspect of subjectivity) confronted with its own unity of all the parts of itself (in the aspect of objectivity) thus is confronted with its own unity as a whole, with itself mediated through the organ, having itself: “as mediated unity, the whole of the actual body is its own means” (ibid:189).
In other words: The organ mediates the self-relatedness of the living thing insofar as in the aspect of being the subject of living it can have itself in the aspect of its physical body through its organs. In a sense, the organism becomes its own organ, its own means of living (ibid:191).
Organs as mediating inner and outer – the Circle of Life
If the organism is to become its own means of living, this doubling cannot happen without the organism – in the aspect of being a whole and the aspect of being a whole in its parts –being set apart from its own unity, gaining distance from itself and coming back to itself (ibid:191–92).
As just shown, the inner self-relation (self-parts) is mediated through the organ(s) – either physical organ(s) or the organism itself – but now the very same mediation (which is so far only understood as an inner one) has to be intuited under the aspect of the relation the organism has to the outside of its physical being that resulted from positionality: the organ, for allowing the organism to be set apart from itself, has to be open (or open itself up) to the space-like relatedness to the positional field outside of the organism (ibid:192).
In other words: The organ has to be a medium for a positional relation to the outside of the organism as well. This opening itself up, going beyond itself into the positional field and back into itself, constitutes a circle of life [Lebenskreis] that runs through organism (core, the whole) and positional field in equal parts (ibid). Organisation as the self-mediation of the unity though its parts thus entails the openness towards the surroundings of the organism (ibid).
Animals as the closed form of organisation
In the context of animals, Plessner offers a new take on positionality that integrates new concepts won by consideration of the form of organisation as well as a new problematic dimension emerging out of it: For a true mediation between the organism and its surroundings, the mediating entities (organs) would have to be in some sense separate from the organism – as its physical body already is in immediate contact with its surroundings – and nevertheless be in relation to it (ibid:227).
Thus, the organic being has to reach a level of being that is different from its physical body comprised of its organs, so that the organs truly become a mediating layer between the living being and its surroundings: this way it is not the whole of the organism itself that mediates the outer relation – which it physically can only do at its contour – like it was the case for plants (ibid).
This falling apart can only take place by organisational means of the physical body of the living being itself, though, has to be a physical reality, has to be an “organisational principle” (ibid:228). Thus, the physical body has to “fall apart” in the sense that “two organizational zones standing in opposition to each other must be created so that the organism can separate into two relatively self-sufficient parts” (ibid:229). These organisational zones are sensory organs on one hand and motoric organs on the other (ibid:230).
They cannot really be separated, though, but must form a relational unity: they are (mostly) mediated as a unity through neuronal centres that are themselves organs and physically and functionally integrate sensory and motoric organs into a single “function-circle” (ibid).
This is how the living being is not in direct contact with its surroundings anymore, but only mediated through the organs: it has contact with its own body only in form of the central organ and is thus set apart from itself insofar as its own physical body in the form of sensory and motoric organs – that which is in direct contact with its surroundings – is only in contact with itself mediated through the central organ. Therefore, animals are by virtue of their closed form of organisation living in a relation of a mediated immediacy with their surroundings (ibid:169-171, 230-231).
Applying the structural characteristics of positionality described earlier, the body that the living being “has” can only be the central organ, it is that which the subject of life has a direct (spacelike) relation with (ibid:231). The central organ becomes a bodily intermediate layer, it is the body that is had and lived – the lived body [Leib] in which the subject of life in turn is positioned – while at the same time it can physically only “be” in and part of the whole of its body:
[The living being] is set apart from itself and dependent on itself as a body. Even in a purely physical way it is “its lived body.” The spacelike center, the core, or the self thus no longer immediately “lies in” the body. To be more precise, the center inhabits a dual spacelike position [Lage] in relation to the body: in it (inasmuch as the whole body including the central organ is not its lived body and does not depend on it) and outside of it (inasmuch as the body depends on the central organ as its lived body). (ibid)
If the lived body is only the central organ that represents the other organs through neuronal signals, it becomes clear that the subject of life does not “have”, is not “ in control of” all of its physical body and that the difference between lived body and (physical) body is a real one: it can be in control only insofar the physical body is represented as noticeable sensory input and motoric output (ibid).
This physical organisation by centres that characterises the closed form thus makes it possible for the structural characteristics of positionality – which demanded the body to be the medium of its relation to the surroundings, i.e. organisation – to become real as an organisational principle of the physical body itself (ibid:232).
The positionality of the closed form together with its self-mediation through the lived body leads to the self or the whole of living being standing in opposition to both its positional field and its physical body. Standing in the middle of its physical body, the physical body and the positional field attain the character of concentricity, where all positions are in relation to the single centre that is the subject of the whole (ibid:239). This means that on one hand, the animal, by its lived body, is able to become aware of itself (in the sense of “its physical body”) and insofar “really” has itself and is a “reflexive self [Selbst] or an itself [Sich]” (ibid:238), but on the other hand, it has no means to become aware of the fact that this awareness is an awareness of itself, as it has no distance to the relation lived body – physical body, but is directly set into this relation, oscillating between the two aspects: the animal lives in an absolute here/now that stands in fundamental opposition, in frontal position or frontality to the surrounding field [Umfeld] it has to take as a given (ibid:238–241).
The peculiar positionality of the human
First, the category "the human" is no empirical category. It is, like the categories of "plant", "decentralistically organised animals", and "centralistically organised animals" a hermeneutical category that tries to account for substantial phenomenal differences (ibid:219,234-35,293). Thus, there is no reason whatsoever to think that it is only humans (as a biological species) that are "human" or "excentrically positioned". To the contrary, if a living being exhibits behaviour that can only be understood if they are excentrically positioned, they can with good reasons be called "human" in this sense (which may or may not have moral implications).
But we would know, as they would be able to share our objective world (worldliness - the taking part in inner, outer, and we-world - is a consequence of excentric positionality).
Now, the lived body of the human (and all centrally organised animals!) is its central nervous system, especially the brain. The human also is an animal in closed organisation, i.e. its surrounding field is principally structured as standing frontally against itself. But as Köhler found out in his experiments with chimpanzees (1917): Humans can make inferences about stuff that is not given in their surrounding field as a given phenomenal content.
We are able to know "what is not there" by differentiating between a) our relation to the surroundings and our body (the absolute standpoint of here/now) and b) a state of affairs bearing possibilities outside our immediate drives and knowledge. Thus, we have to stand in a relation to that, have to gain "distance" - positionally, not physically - and be posited "behind" our bodily existence (both as a lived and a physical body), into nothingness.
Only then, we can understand why we are able to set ourselves outside of our relation to the surroundings, contemplate nothingness, have culture as a natural aspect of our being.
The characterisation of excentric positionality itself is much better described by Marjorie Greene in the answer of @GeoffreyThomas, though.
Positionality is a spacelike relation between the subject of life and the physical body through which it lives. Excentric positionality is the try to render the fact that humans are able to understand themselves and their surroundings as a state of affairs, as a space of things of potentialities, of what is not there but might be, i.e. cultural and ethical beings.
This cannot be understood if they are only living beings insofar they are a physical being with a natural place, i.e. an animal: This would mean they are in frontal opposition to their surroundings, i.e. can only see and act upon what appears to them as corresponding to their bodily or psychological needs. It then serves as a means to explain all behavioural expressions that are specifically human:
Essentially, excentric positionality - something that Plessner ultimately derives as a characteristic of the human being as a living being that is part of living nature and justifies phenomenologically and empirically - allows us to understand that the Cartesian dualism (and radical scepticism) is wrong, i.e. that, how, and why we are justified in assuming that we really have access to "essential being".
At the same time, it allows us to understand all human behavioural expression as emerging out of excentric positionality, something he catches in the three fundamental laws of anthropology: natural artificiality (explains culture, language and its diversity), immediate mediacy (here the epistemological aspect is caught - explains knowledge), and utopian standpoint (ethically and historically - explains religion, morals, and historical relativity).