I just didn't get it. I'm not totally sure I do get it yet - but I've kinda convinced myself of twitter by just thinking of it as an open IRC where you subscribe to people rather than chat rooms.
To use messaging middleware speak, I've not been the biggest fan of IM as it tends to lead to loads of point-to-point conversations with lots of duplication. I find myself saying the same thing again and again to people on IM. Plus IM feels way too invasive; I tend to hide away from IM to avoid getting hassled these days :). With IRC I can say stuff to the room instead which avoids me having to say the same thing again. Plus IRC tends to feel less invasive; I can read it when I feel like it but otherwise kinda ignore it
Twitter so far is kinda feeling like a combination of blogs, IRC and SMS; where you subscribe to people rather than rooms which helps cut down the noise. If someone's getting annoying or boring, just stop following them. The brevity of posts (140 characters) also helps trim down the noise and fluff leaving a mostly interesting stream of kinda mini-blog-posts and banter. So far I'm hooked! :)
If you are on OS X try out twitterific as a Twitter client. While IM is a pretty reasonable way to work with twitter - I do like not running my IM client and using twitterific instead :)
If you have an iphone there's a bunch of clients available; so far I'm liking twinkle