"can't think of a legitimate reason why anyone would want to. "
Because everyone has a right to privacy, regardless of the reason why?
An IP address is something like 72.14.207.99 (that's a google address).
It just specifies where your computer is on the network.
If you want to hide your IP address, you can use something like Tor (The Onion Router). What this does is use a set of other people's computers to route through, so that your starting IP address isn't revealed as easy.
It's not perfect, but helps with anonymity.
http://www.torproject.org/