I've had a few conversations with friends and colleagues about Twitter lately, some of whom feel "it's stupid" or they don't really get it. My response is that the mistake is to think that Twitter is just for one thing. It's what you make of it: A microblog, a RSS feed for your blog, a news feed for you, a social outlet to share experiences (like the inauguration or Oscars), a communication tool with other bloggers, networking with people in your field, etc.
However, I'm not sure that answer would do much for Jon Stewart.
Of course, Twitter has that sort of "trendy" vibe, and I think that's what puts some people off of it. And when politicians, journalists, and broadcasters adopt it to look cool, doesn't that make it the opposite of cool? (I do believe, however, that some of those people are sincere about using as a communication tool.) But maybe I'm the wrong guy to talk to about this, because I love Twitter.
By the way, if you really are curious about Twitter and looking for a good explanation, I recommend two videos that Steve Isaacs posted as sort of a primer. (His vlog is worth a regular stop, anyway. Unlike, say, Samantha Bee's pretend one.)