Your broadband is split off from your phone line (PSTN) at your local telephone exchange in much the same way as they are separated in your home by the use of a microfilter or "accelerator" box.
The phone connection is then handled by your local telephone exchange less than three miles away usually, and your broadband connection is "piped"off to a completely different switching centre designed to handle that.
The centre for handling your broadband may be many miles away, even a hundred or so, and may also change occasionally as traffic requirements alter.
Your "public" IP, if you have not paid for a "static" one from your ISP which stays the same always and is mainly used because a server needs a permanent address for incoming access, will be a "dynamic" one which is always allocated by your ISP and may be changed as required, and neither you or your computer have any control over it at all.
The multi-page article in the link below gives you a basic description and block diagrams of the way it works;
http://www.kitz.co.uk/adsl/equip.htm
And this is what the equipment actually looks like;
http://www.kitz.co.uk/adsl/telephone_exchange.htm
That`s about as simply as it can be described I`m afraid.
Regards, Bob.