Transmission stores your torrent files in your standard config folder, at
~/.config/transmission/torrents where ~ is your home folder.
The .config folder is usually hidden. To get there, open the File Browser and go to your home folder. Choose View > Show Hidden Files in the menu, then look for ".config". In that folder you'll find transmission, and the rest should be pretty straight-forward.
(By the way, to find a file in a long list you can just start typing its name).
The torrent folder contains every current torrent, and it includes torrents downloaded from magnet links as well. You can just open those with another client and it will work right away.
Remember to turn Show Hidden Files back off later on, or you'll see a lot of strange looking files: