For decades there have been a wide variety of shady filehosts that will happily host content with no regard for IP and offer downloading for the same (good for them). They manage to make money by offering "premium" subscriptions that allow to download without having to wait / bandwidth limitation (these days you even have services that try to mutualize such premium accounts between users for a smaller fee, using their proxy to serve their own users). For just as long there have been websites that index those direct filehost links, and make money through either ads or members donations. It's an alternative to torrenting. Gog-games is an example of such an indexing website (there are many, many others). 1fichier is an example of the filehosters I mentioned above (same remark).

To answer your question, the reason they don't go down is they routinely operate in jurisdictions that are hard to act on by LE in the imperial core; they also often pay lip service to DMCA requests by actually removing content after reports, though they'll almost universally make the process complicated, long, and pretty useless (not removing identical files reachable from other links, for example).