How do magnet links make them harder to shutdown? Is there a significant legal benefit to them over torrent files, given that as a user the two options function nearly identically.

They're using DHT, a "distributed hash table". Basically this means you can find "peers" (other people that have or want the data for a particular torrent) without needing a centralised tracker. This means if TPB goes down, everyone's torrents keep on working.

Now the magnet links mean you don't even need to download a torrent file, you JUST need that magic hash, which uses the DHT to find the torrent metadata itself, as well as the data. Neat! All you need to disseminate data to other people is to give them that tiny hash, you don't even need a website for that - you could spray paint a QR code somewhere.

The last part of the equation is that anyone can download a dump of The Pirate Bay to put up their own clone, so even if the canonical site does get taken down, all the torrents will live on.

Magnet links take up much less space and it's much easier to mirror TPB