From a modern perspective, we are accustomed to distinguishing between the way the universe is, and our experience of it. The universe is complex and weird and we struggle to understand it and navigate our way through it. Our thoughts and beliefs are an attempt to make sense of our experiences. Our theories and even our logic are an attempt to organise and systematise our experiences and render them into a form that has predictive and explanatory value.
But this is not how ancient philosophers saw things. They imagined a kind of direct correspondence between thought and the way the world is. The world contains true propositions and the job of a rational person is to believe these.
On the modern view, it does not make sense to say that the universe 'contains' contradictions. Contradictions are a feature of our understanding and our language, not a property of the universe. The universe just does what it does. Contradictions arise only when we attempt to describe it. In the most commonly used logics, including classical logic, no contradictions are true. This is not a statement about how the universe is, but a statement about how we choose to organise our information. We typically find it inconvenient and unsatisfactory to allow that contradictary propositions are true.
On the ancient view, a contradiction would mean that the world has some conflicting and impossible combination of properties. The ancient Greeks, at least, held that the world is rational and so this was assumed to be unthinkable.
Kant can perhaps be thought of as the dividing line between these positions. Kant held that the world as the thing-in-itself is unknowable. But he held that because we understand the world in particular ways (the categories) we can formulate transcendental arguments about the way the world must be in order for us to have any knowledge of it. For Kant, this gives rise to a priori knowledge, including logic.
As to Aristotle himself, as the article from SEP shows, Aristotle presents three different versions of the principle of non-contradiction. (1) He says that it is impossible for a thing both to have a particular property and also not to have it, at the same time and in the same respect. (2) He says that it is impossible to believe a thing both to be and not to be. (3) He says that opposite assertions cannot be true at the same time.
All of these are different, and we would express them differently in modern logic, and it is not clear whether Aristotle regarded them as distinct, and if so, how they are related. In the way that Aristotle defends the principle with his elenctic argument, he seems to be leaning on the claim that reasoning and belief would be incoherent without it. This might be considered similar to a transcendental argument since it is not so much holding that contradictions are impossible, only that we would be unable to think and communicate rationally if we believed contradictions. And since we are capable of thinking and communicating rationally, contradictions must be rejected as false.
Switching back to the modern perspective again, this argument could be said to be question-begging. Why should we suppose that any and all contradictions completely subvert rational belief and communication? People do seem to be able to cope with inconsistent beliefs without becoming completely dysfunctional. Dialetheists, while being in a small minority, allow that some contradictions are true without triviality and without explosion.
