Simle answer: he doesn't, any more than stage magicians make bunnies disappear. He just creates the illusion that he has mastery over the topic of the debate. He looks impressive, but that is the benefit from being able to get all of one's facts from Noam Chomskyland. The rest of us mere mortals have to do research, and rely on sources that are not just ourselves (if you've ever read a Noam Chomsky paper, you would know what I mean. If the man cites references at all, 9 times out of 10, it is to a paper that he wrote. In the 10% of instances when the reference is not to himself, 9 times out of ten, it is to someone who has been his lackey for a while). This creates a situation where Noam emphatically and passionately lectures the audience about a world that exists in his head, and his opponent is forced to do what he can with a world that has to be objectively verified.
Additionally, he's a big fan of ad hominem attacks on his opponent, which are compelling, but illogical, particularly when they cannot be substantiated or warranted in the debate (I've already pointed out that the Noam is really not a fan of substantaitable claims). He's a lot less impressive if you do follow up fact-checking research after a debate. There is also the part where he comes off as deeply profound by spurting out strange 'factoids' that definately have no correspondence or realistic consequence in a ponderous tone, typically using big words and conflating concepts that have no place being together (eg, his perrenial argument that the demand that Israel be recognized as a state is the same thing as insisting that Palestine not be...this is only true if you take the 'with me or against me' line that most anti-Israel lectures tend to take--projecting their bigotry onto Israel) He does the same thing in linguistics (my field of study).
To correct another poster: I think that Halle did more for GTs than Chomsky (he's the one credited with delivering the killing blow to Structuralism), and John McCarthy did much more to revolutionize mordern linguistics with his work on nonconcatenative morphology and his expansion of optamality theory. I don't know why the Noam is as adored as he is in psycholinguistics--it hardly takes a sleuth to point out that Skinner's book was moronic, and he generally opposes (usually by misrepresenting) theories that are supported by cognative research (optamality theory, especially). I think that psycholinguists blind adoration of the man has severely impaired their field (for example, it's only in the past 10-20 years that any research has been done on multilingual children). Many other theories of linguistics (lexical phonology, most famously), were created just to keep the Noam from being too academically irresponsible (there is no such thing as an underlying velar fricative in an English word, and Japanese is a real language, exc...). Noam just makes a lot of noise---he's probably done a lot of damage to the field, by disparaging field research (basically, all his theories belie the belief that every human language is secretly English...and not just English, some wierd dialect of American English that is in his head and has grammatical rules that can convieniently change to reflect any point that the Noam wants to make...) and discourse linguistics. The field is only now starting to be more accepting of cultural roles in the development of languages.
I'm not being libelious--you asked for my opinion, and I gave it, based on my experiences in my field, reading his works, and in watching him "debate". I already provided you with ample theories that Chomsky opposes that have revolutionized modern linguistics, and evidence that he is academically irresponsible (ie, proposing an underlying velar fricative for some English words, simply to force them to fit his theories, like a troublesome puzzlepieces). If this were a debate, I would come with more complete citations...including a very rude e-mail he sent to one of my professors who asked him for some clarification on one of his ideas. His responses to theories he doesn't like is to misrepresent them, then say that they are stupid. For example, his reaction to OT theories was to conflate various proposed cognative mechanisms, and then say that the 'theory' made no sense. Indeed, OT theories that have those proposed mechanisms conflated do not make sense---that is why traditional OT theories do not conflate them.
I am a graduate student in linguistics (one who actually does field research and will be expected to do more than cite hypothetical varieties of "American English" in my dissertation)--which means that I have taken quite a few classes on theoretical linguistics. I can't speak for all philosophy, but I can happily second your claim that he has not revolutionized my field. Nowadays, most linguists ignore him, except for older American ones--typically on the east coast. I have read many of his papers trying to smear other theories of linguistics---neo-Wharvian ones (even though I am the first one to admit that Daniel Everett of Piraha-number fame is an obstinant asshole himself--but he's a brilliant obstinant asshole who bases his work on an actual language and actively invites other academics to come down to the Amazon and check out his theories), Optimality Theory, and field research in general. Typically in a university where professors do actual field research, especially on tiny, dying languages, you will hardly find anyone who really likes Noam Chomsky. Talk about him in countries where virtually all linguists ARE field researchers, and you would get a flat stare. And yes, MIT is known as the Generative stronghold of linguistics, trapped in the 60s by Chomsky's grasp. The only linguist that I can cite off the top of my head who came from there and was not willing to be a Chomsky flunkie is John McCarthy, who I have already cited as amazing (largely, because he is able to recognize where his theories need to be tweaked).
Additionally---I don't know why mathematicians would like the Noam--I have yet to meet a comp sci grad who doesn't hate the guy.