The reason bittorrent is more popular than rapidshare is that bittorrent does not require a server to back any files.
bittorrent works by sharing parts of the file instead of the entire file one big chunk at a time. You can connect to thousands of users and download tiny pieces from each user even if that user does not have the entire file.
Also because bittorrent splits the main file into thousands of little pieces, the download can be stopped and started as much as you want without having to worry about connection to any serer.
i know that's a bit of a confusing description of how bittorrent works but you can actually learn all about bittorrent from wikipedia: