You need to download and install a 'bittorrent' client
Try this one:
http://www.utorrent.com/
When you click on a torrent download link, it downloads a small torrent file which manages the download by connecting all the bits of the file to the places/users it is available from at that time.
Over the time of the download, the actual file (like mp3 or whatever) is built up so when its done, its just a case of opening that file.
Thats pretty much it.
