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).
Indeed--it's so striking that most people don't even realize how much he does it unless they look at a Chomsky 'works cited' page themselves.