When you download a .torrent file, you're essentially downloading a small file that contains information on the larger files you want to download. The torrent file tells your torrent client the names of the files being shared, a URL for the tracker, and more. Your torrent client then calculates a hash code, which is a unique code that only that torrent has—kind of like an ISBN or catalog number. From there, it can use that code to find others uploading those files, so you can download from them.
A magnet link does away with the middleman. A magnet link is essentially a hyperlink containing the hash code for that torrent, which your torrent client can immediately use to start finding people sharing those files. Magnet links don't require a tracker (since it uses DHT, which you can read more about here), nor does it require you to download a separate file before starting the download, which is convenient.
Magnet links are dead simple to use. Just click on the magnet link, and your browser should automatically open up your default BitTorrent client and start downloading. It's that easy.