Welcome to Transmission!
Transmission is a BitTorrent client. It is used for peer to peer filesharing over the internet. BitTorrent is a fast way of transferring files, because everyone who downloads must also upload to others. This means speeds are only limited by the number of people connected to a given torrent - the more people the better!
How do I use it?
Download your file's associated 'torrent file' (extension .torrent). These are commonly found at 'tracker' websites.
Once you have the torrent file, click Open and select it, or simply drag it into Transmission - downloading should start immediately.
Some sources provide the so-called 'magnet links' in order to download torrents. Transmission supports magnet links but you will probably have to associate them with Transmission from within your browser first. Alternatively, you can select File > Open URL... to manually add a magnet link.
You can pause and resume transfers at any time, so long as the files remain in your download folder.
It is good etiquette to share or 'seed' the file for a while (ie leave it uploading to others) once your download is complete. That's how you downloaded the files in the first place!
You can use a torrent's Properties window to adjust download/upload preferences on the fly.