It gives them a way to quickly communicate feelings, and to see what other people are saying quickly. Because it's so short, you can follow a lot of news without being overwhelmed.
Async communication is key to twitter. It's also cool that you can tweet by any method anyone has ever thought of, from almost any device that can send any sort of data whatsoever. The API has helped them greatly. I'm sure that someone, somewhere has a computer that transcribes short voice messages that can be left by phone. If not they will by the time I hit "reply" ... there should be a rule #34 for twitter.
* e-mail
* sms (key for reaching mobile people w/o smartphones)
* web browsers
* native apps / widgets
And then services like twitpic built around twitter.
