People generally like to know how they are doing, and a few quick measurements (post count, post rating, Thanks given) here on the forums are given in numbers, so it's natural for people to pay attention to those numbers. Out of the three, only post count is public, so it provides a quick way for anyone to see where they stand in a crowd - I suspect many people who track their post count for this reason see it at least in part as a status thing. On the other hand, some people put a lot of time and work into each of their posts, especially if they're writing message board solutions for an online class, solutions for problems on the forum, or other thoughtful responses. In this case, post count can be a rough estimate of how much good work they've done.

In terms of "caring about your own post count," I think there's a huge difference between "letting a desire for a higher post count motivate most of your writing on the forums" and "having a rough estimate of what your numbers are but otherwise not worrying about them." I think most of the people in the latter category, myself included, realize post quality (tone and conciseness included) are way more important than post count itself.