Well, imagine trying to communicate with someone and you expect an answer, but they don't know who to reply to.
The ONLY reason someone would really want to hide their IP is if they're a
Cracker - evil hacker - and they want to "spoof" their address to crack another system.
And this is possible ... but if you want to know how to do it, you'll just have to figure it out ...
As far as changing IP's with dial up/cable/dsl - you're getting an address from a Dynamic Host Configuration Protocol server. When you release it, your system is telling the server you don't need the address anymore and it can free up the resource (because it has a limited number of them). If you renew, immediately, the server has the ability to remember your MAC address and will likely give you the same address if it hasn't been leased to someone else.
If you have an always on connection - such as cable or dsl - your side will begin contacting the server to renew the address when it is half way through the lease period. If it can contact the server, it will renew and then wait for half the renewed period. If not, it will keep trying periodically.
HTH