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Max Tegmark believes the universe to be a mathematical structure, and he further claims any mathematical structure with self-aware substructure will perceive itself in a physical world. What exactly does he mean by self-aware substructure?
Is he employing structure in the same sense we generally use in mathematics: like, for instance, algebraic structure, differential structure, etc.?
Here are some quotes from Max Tegmark's "The Multiverse Hierarchy" hopefully answering the OP's question: What exactly does he mean by self-aware substructure?
Tegmark assumes that "all aspects" of reality are isomorphic to a mathematical structure.
Let us now digest the idea that physical world (specifically, the Level III multiverse) is a mathematical structure. Although traditionally taken for granted by many theoretical physicists, this is a deep and far-reaching notion. It means that mathematical equations describe not merely some limited aspects of the physical world, but all aspects of it. It means that there is some mathematical structure that is what mathematicians call isomorphic (and hence equivalent) to our physical world, with each physical entity having a unique counterpart in the mathematical structure and vice versa. (page 10)
We are part of reality and so we are isomorphic to a part of this mathematical structure. That makes us a substructure of it. Since we are self-aware, we are a self-aware substructure (SAS).
Given a mathematical structure, we will say that it has physical existence if any self-aware substructure (SAS) within it subjectively, from its frog perspective, perceives itself as living in a physically real world. What would, mathematically, such an SAS be like? In the classical physics example above, an SAS such as you would be a tube through spacetime, a thick version of what Einstein referred to as a world-line. The location of the tube would specify your position in space at different times. Within the tube, the fields would exhibit certain complex behavior, corresponding to storing and processing information about the field-values in the surroundings, and at each position along the tube, these processes would give rise to the familiar but mysterious sensation of self-awareness.From its frog perspective, the SAS would perceive this one-dimensional string of perceptions along the tube as passage of time. (page 11)
The multiverse idea allows him to use the anthropic principle to claim that there must be at least one of those universes able to allow for human life since we exist.
...although many if not most mathematical structures are likely to be dead and devoid of SASs, failing to provide the complexity, stability and predictability that SASs require, we of course expect to find with 100% probability that we inhabit a mathematical structure capable of supporting life. Because of this selection effect, the answer to the question “what is it that breathes fire into the equations and makes a universe for them to describe?” (Hawking 1993) would then be “you, the SAS”. (page 13)
What Tegmark means by a self-aware substructure is self-aware reality like us. Since he believes all aspects of reality is a mathematical structure this requires him to think of us as substructures of that mathematical structure.
Tegmark, M. (2009). The multiverse hierarchy. arXiv preprint arXiv:0905.1283. Retrieved on August 24, 2019 from arXiv.org at https://arxiv.org/pdf/0905.1283.pdf
If Max Tegmark is using the words "self-aware substructure of the universe", then he is referring to agency and the beings who possess it, for example us. If the universe is perceived as a mathematical structure, than not all of the structure possess the capacity for perception, hence the delineation of a sub-structure which by definition means a smaller portion of the structure.
Is he using structure to describe physical reality the same way he is to describe, say a non-Abelian group? Yes. Does it make sense? To some extent yes. Imagine being lost in your head and thinking that all of your thoughts are information, and that all of the things your thoughts reflect can be thought of as information, for instance particles are described as mathematical waves, or atoms can be described with bits, then it might be easy to just declare the universe itself mathematical instead of separating the universe into an external, concrete reality made of matter, and an internal, abstract structure made of logical, arithmetic, and set theoretic structure. Many bright people think along these lines.
Of course, those of us who are realists of various flavors think this thinking is silly. Rocks may be made up atoms, but our thoughts which follow from the computational functionality of the neurons in our head are not the same as the rocks we think about. This is a theme that Platonic roots where numbers are instantiations of Forms. A famous example of this thinking is also encompassed by an essay called The Unreasonable Effectiveness of Mathematics.
There are plenty of arguments against a Platonic view of the universe, but alas, you didn't ask about them. ;) But do consider this. Note all messages require a medium. If you deny this, identify a message without a physical medium. Now, if all messages, or all information requires a physical embodiment, be it electrons in a flip-flop or clay or paper and ink, then the question stands, what is the mathematical structure of the universe expressed in? What is the medium that encodes the information that represents physical reality? If you are unable to provide scientific proof that this medium exists, then you must reject the notion that the universe is made of information until you can meet empirical requirements. If you reject the need for empirical requirements, then you are participating in pure rationality and Metaphysics with a capital M, which by scientific thinkers is generally labeled as meaningless. So, you find yourself having to choose between science or the position that the universe is made of information.
Now ask yourself, are you willing to bet against science? If you are, then you might consider argument by defenestration, and prove us realists wrong and throw yourself out a window to prove us wrong. I would advise against it, of course.
EDIT 2019-08-26: In response to the fair critique that the argument against Tegmark's claim is circular, first, remember that Tegmark is the original claimant, and as such, the burden of proof to show that the universe is a mathematical structure would be his to show that the universe is a mathematical structure (presumably represented in some sort of math machine). Obviously, his actual argument isn't present here, so I've outlined a sketch of an objection.
Is the argument circular, as suggested? As a skeptic and empiricist, in a way, yes, I have rejected the notion that the universe isn't a Platonic realm as an assumption, and hold that the burden of proof is to show that it is. Perhaps Tegmark has an explanation of how everything being a mathematical structure allows me to have sensory input to discriminate the material from the non-material, presumably as some sort of simulation. So, yes my manner of justification presumes some elements of my conclusion, but that's ultimately because rationality has limits, and it is those limits that show that a purely rational answer cannot be the solution, and that we must appeal to our physical embodiment as a method of discriminating what is, and what is not. See my response in the post regarding the Muenchhausen Trilemma
By "self-aware substructure" he means a connected region of spacetime, such as your worldline, that contains the physical processes that supposedly give rise to consciousness. He further describes it as a braidlike bundle of particle worldlines that exists for a bounded time interval, beginning some time after your conception and ending at your death.
