Just open the torrent file with a BT client.
The official BT client:
http://bittorrent.com/
An advanced BT client:
http://azureus.sourceforge.net/
If you're getting large files (>500megs or so), lots of smaller files simultaneously, or helping to seed a torrent, you'll want Azureus. It gives you a lot more control over upload and download rates, which helps to maximize your internet connection.
If you're just downloading stuff occasionally, the official client is super-simple to use. (Doubleclick the .torrent, and choose where to save. Then watch the progress bar
)
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The .torrent file itself tells your computer where to look for the downloading file(s), how to doublecheck if a piece fits, and so on.
The BT client then connects to the seeds (those with the whole file, servers basically) and other downloaders. You trade pieces of the file you've already got for pieces you need, and everybody gets a faster download as a result.
Note: Only pieces of the file you are downloading get traded, not anything else.
Each piece you send and recieve is doublechecked against the corresponding .torrent file to make sure it isn't corrupted.