I 'got' Twitter pretty quickly, because I was actually using it before it existed. I used to carry on lossy low-priority conversations with people via AIM status messages; it was a good way to say stuff that wasn't important enough to warrant a message and also to broadcast things in the vein of "hey, here's a fun link" without needing to directly spam people.
I'm pretty sure this is roughly what the designers of Twitter had in mind, incidentally. If you take a look at the XML or JSON feeds provided by the Twitter API, each individual tweet is actually called a 'status' — the URI for a particular tweet is
<username>/status/<some_big_number> — the
<OL> element that contains the tweets on Twitter HTML page is of CSS class 'statuses'... the list goes on. Obviously the original intent of the tool doesn't define what the tool is actually used for — I don't think AIM away messages were intended as an out-of-band communications protocol, at least at first — but it's interesting nonetheless.
What's also interesting is that Twitter users seem to fall into two separate and largely non-overlapping groups. Group A are people using it much like I am, as a non-ACKed broadcast medium for random thoughts that aren't important enough for a reliable messaging protocol like IM, email, or a blog post. Group B are the people who see Twitter as a marketing/communication tool that will allow them to get the word out to members of Group A, for whatever value of 'the word' is meaningful to them.
I see three ways this can go: either the two groups diverge completely and group B develops a self-sustaining ecosystem of their own; or group B recognizes that marketing in this context is futile and dwindles and dies; or group B is so persistent in their efforts that group A abandons Twitter and finds a new bicycle.
If you get the 'new bicycle' thing, you're definitely in group A, though not getting it doesn't necessarily relegate you to group B.
But if you know you're in group B and want to understand group A, then you should figure out what it means, why you don't get it, and what you can do to make sure you get to autotune the new new bicycle... and then despair of ever marketing effectively on something like Twitter.