There was a Question on Superuser asking how to set up a torrent server.
(The answer was that there's no such thing as a torrent server.)
We're not talking about a torrent server here, but the principles of sharing by P2P software will still apply.
When people install P2P software on their computers, they share files between themselves with no server involved.
Each user starts by downloading a relatively small torrent file which specifies a shared file. The P2P software allows them to download pieces of the shared file from the other users, and importantly share the pieces they already have with further users (by uploading).
The longer you stay connected to the peers, the more pieces of the file you have, and ultimately, of course, you're going to stay connected until you have all of the file.
There is a potential to share all of the file specified in the torrent. If you have downloaded all the shared file, there's a good chance you will have uploaded the entire file to someone else.
That's why it's called sharing.