This has been a particularly difficult section for me. Please forgive me if things sound opaque, but Kant is a particularly difficult writer and my reading is hurt by the fact that it isn't occurring in an institutional setting. But enough disclaimers.
I'll start by backing up a bit and providing my reading of the preceding few sections, namely a summary of the transcendental deduction, the explanation and outline of book two, The Analytic of Principles, and the ensuing introduction. Against this background I'll present my trouble with chapter one, Of the Schematism of the Pure Concepts of the Understanding.
We've just completed the Analytic of Concepts, where we were informed of the necessity that the Pure Concepts of the Understanding strictly refer to sensible intuitions, thereby composing possible (which is to say sensible) experience, owing to the elementary unity of original apperception. Put differently, if it is the case that your particular experience of the world uniquely belongs to your singular, and therefore unitary, consciousness, then it must be the case that experience is composed by some kind of knowledge about objects prior to any actual experience. Or another formulation: if it is possible for me to say that my own or another person's thoughts or perceptions are subjective, then it has to be the case that the structure of subjectivity is objective, and it is this precise structure that Kant illustrates for us in the Analytic of Concepts, specifically the transcendental deduction, and which objectively provides the conditions of possibility for experience, Kant defining experience as determinate knowledge of appearances (I think I needed a semicolon somewhere in there but I'm not sure; also, please free to correct me where you think my readings are inaccurate).
So now Kant moves to Book II, the Analytic of Principles. The purpose now is to explain just how the pure concepts of the understanding are applied to or provide rules for determining appearances, for so far he has only described the necessity that they refer to sensuous objects by way of the transcendental unity of original apperception, the bare fact of consciousness. Judgement is the faculty tasked with "distinguishing whether something falls under a given rule or not." Transcendental logic can assist here by pointing out "the case to which each rule is to be applied."(B175, A136) So the first chapter of Book II "will treat of the sensible condition under which alone pure concepts of the understanding can be used, that is, of the schematism of pure understanding."(B175, A136) Still following? To recap, given that we possess pure, a priori concepts that necessarily and strictly refer to sensible objects, how does a non-empirical and non-sensible thing like a pure concept interact with a sensible object?
This where things get weird.
Kant tells us in the first chapter that, given the heterogeneity of pure concepts and sensible intuitions, there must be some third thing which mediates their interaction. This thing would have to be, on the hand, pure and non-empirical, but on the other hand intellectual and sensible at the same time. Kant titles the thing that satisfies these conditions a "transcendental schema". But just what is a schema? Kant replies:
After what has been said in the deduction of the categories, we hope that nobody will hesitate in answering the question whether these pure concepts of the understanding allow only of an empirical or also of a transcendental use; that is, whether, as conditions of a possible experience, they refer a priori only to appearances, or whether, as conditions of the possibility of things in general, they may be extended to objects in themselves (without any restriction to our sensibility). For there we saw that concepts are quite impossible, and cannot have any meaning unless there be an object given for them or, at least, for the elements of which they consist; and that therefore they cannot apply to things in themselves (without regard as to whether and how things are given to us). We likewise saw that the only way in which objects can be given to us is by modification of our sensibility; and lastly, that pure a priori concepts, besides the function of the understanding in the category itself, must also a priori contain formal conditions of sensibility (particularly of the inner sense), namely, conditions which contain the general condition under which alone the category can be applied to any object. We shall call this formal and pure condition of sensibility, to which the concept of the understanding is restricted in its use, the schema of the concept; and the procedure of the understanding with these schemata we shall call the schematism of the pure understanding.
The schema in itself is always only a product of the imagination; but as the synthesis of the imagination does not aim at a single intuition, but only at the unity in the determination of sensibility, the schema ought to be distinguished from the image. Thus, if I place five points one after the other, . . . . ., this is an image of the number five. If, on the contrary, I think a number in general, whether it be five or a hundred, this thought is rather the representation of a method of representing a multitude (for instance a thousand) in an image in accordance with a certain concept than the image itself. Indeed, in the case of a thousand I could hardly take in the image and compare it with the concept. This representation of a general procedure of the imagination by which a concept receives its image, I call the schema of this concept.
[...]The schema of a pure concept of the understanding, on the contrary, is something which can never be brought into any image whatsoever, for it is nothing but the pure synthesis determined by a rule of unity according to concepts in general, a synthesis expressed by the category; it is a transcendental product of the imagination, a product which concerns the determination of inner sense in general, according to the conditions of its form (time), with respect to all representations, insofar as these representations are to be joined a priori in one concept, in accordance with the unity of apperception. (B178 - B182, A139 - A142)
Now, the key to my reading of all this, which is what is in question after all, lies in nature of inner sense. Kant tells us earlier that inner sense lacks content but not the form of sensibility in general, which is time. Space is the condition of external sensibility but all intuitions are subject to the condition of time. It is owing to this strange situation of inner sense that Kant controversially claims that our self-image, or how we see ourselves, is a fabrication: your inner sense is empty, but still receptive of sensation. But its a law of the understanding that anything sensibly intuited has to be combined into a unitary appearance. So the understanding fabricates the unity of your self-image in order to make-up for this strange shortcoming of inner sense. But the same operation appears again in the context of the schemata, but what subtly distinct way? Inner sense is receptive, but possesses only the formal condition of time as its whole sensibility (which is receptivity). Time, however, turns out to fulfill the general condition of a schema: it is pure and non-empirical since it belongs to transcendental aesthetic as an a priori condition of sensibility, and so is not only intellectual but also sensible as the element of all sensation generally. Subsequently Kant will explain that the schemata are nothing "a priori determinations of time according to rules" but remember that the understanding can't provide such without provocation on the part of sensibility.
So could it be that Kant answers the question "how do pure concepts interact with sensible intutions?" in the following way?: Because we sense perturbations in time, which is purely formal, and time is primarily registered by inner sense, which is empty of content, the schemata are the way our minds cohere (or unify) the manifold of this kind of intuition, these perturbations in time on the part of external objects (distinct from their peturbations in space); that is, our sense of time is the mediating plane between sensations and concepts.
Do I have this correctly? Can anyone suggest corrections to my reading? This was probably overly long and could've been shorter, but I'm not really sure where I would've even begun making that kind of edit (some pointers would be very helpful, thank you). I guess I leave it up to y'all now. I hope I make some kind of sense.