I am pretty new to Twitter, but I must say I get it. It is for people like me that have short attention spans and tend to have little to say. LOL!
Actually, Twitter is great for keeping up with the "little things" in your friend's lives. You post short, one or two sentence, posts about anything you want people to know. It is intended to fill the gaps between meetings, phone calls and blog posts.
It is also a great way to connect with new people. I follow several people living in my area, working in my field and/or just sounds interesting to me. It is easy to start and stop following people, so I start following almost anyone I want to and stop if I don't enjoy their posts.
To lean more about Twitter check out this video: Video: Twitter in Plain English.