Pete Mandik has a post summarizing transcendental arguments. As he summarizes them, transcendental argument has, in terms of its premises,
(1) "an allegedly obvious claim about EXPERIENCE, KNOWLEDGE, or some other feature of one’s own mind"
(2) "a claim about a necessary condition on the truth of the allegedly obvious claim in (1)"
And then concludes that the necessary condition in (2) is satisfied (because, of course, that which requires it obtains).
This isn't a bad attempt, but I'm not sure this is quite right. Putting it this way would make a transcendental argument an ordinary causal argument (taking 'causal' in a reasonably broad sense), but I don't think this characterizes what people are really trying to offer in transcendental arguments. The problem is the "allegedly obvious". I don't believe an allegedly obvious claim is needed for a transcendental argument; likewise, I don't think every allegedly obvious claim combined with a claim about its necessary condition gives you a transcendental argument.
When Kant gives a transcendental argument, he is not alleging that something is obvious; he is alleging that something is unavoidable. This, I think, is the key to understanding what a transcendental argument is, and this alone can really show why transcendental arguments have at least the potential to counter skeptical arguments with some bite. Transcendental arguments have force against skepticism precisely because they are not dogmatic -- it is precisely because they don't use claims because they are 'obvious'. If they were dogmatic they would be completely useless against the skeptic because they would beg the question; the skeptic could merely deny the obviousness of the claim. But the Achilles' heel of skepticism throughout the ages has always been practice. Skeptics themselves cannot get away from practical life. And transcendental arguments are ways of building on this fact, by starting with things that even skeptics can't genuinely avoid doing, that even skeptics must admit they do. So, for instance, skeptics can't avoid making some sort of judgment about the temporal order of their experiences; we can't have such judgments without presupposing independent substances that endure while things change, Kant argues, so we are all already mentally committed to the existence of enduring substances independent of our minds. The foundational claim need not be obvious at all -- that's a secondary issue -- but it does need to be somehow unavoidable in light of what we actually do or are able to do. It's a claim that, given that we do (or are able to do) something, we would be engaged in a sort of performative inconsistency to do (or be able to do) it and not accept what it is identified as a necessary condition for that thing.
Part of the thing that has to be remembered is that transcendental arguments arise in a context where one has already made significant concession to skepticism; it merely hems in the skeptic, so to speak, by identifying things to which the skeptic is committed as a matter of life (usually but not always mental life) and that is really what it is designed to do. In Kantian terms, it is critical rather than dogmatic; it opposes the skeptic as a critical and not as a dogmatic argument. It doesn't give you reality, but it isn't supposed to do that: it's supposed to give you what you are committed to in living a mental life, whether you are skeptical or not. As an anti-skeptical tactic it is not skepticism-destroying but skepticism-bounding; it limits rather than refuting by establishing a way in which the claim is a lawful or legitimate claim, a claim we have some right to make -- a right the skeptic (also committed in practice to the claim, even if they suspend judgment about whether it is really true) cannot really deny.
