Though we're yet to really go deep into Hell Is Us's awfully New Weird narrative, it's plainly obvious at a glance that this isn't your grandpappy's action RPG. There's a whole bunch of (largely) inexplicable stuff happening in this game, and it shouldn't come as a huge surprise that time flubbery plays a part, too. Indeed, even the game's first proper exploration area, the Acasa Marshes, has an ominous Timeloop smack-dab in the middle of it. What does it do, though?
Hell Is Us's Timeloops are basically extremely potent temporal artifacts that are more-or-less the ingress points for the Demons' invasion of Hadea. In gameplay terms, closing these Timeloops makes it so that the Demons don't respawn, which is an obviously handy thing to do!

Though you might expect a Timeloop to be a simple combat gauntlet in Hell Is Us, this couldn't be further from the truth. Well, okay, the bits you need to do before heading into the resident Timeloop sure do involve a lot of combat, but that's beside the point. Here's exactly what you need to do to close a Timeloop:
Simple, right? Not at all! There's a lot of intermediate steps to go through before you can do any of this, and the Watcher's Nest puzzle is an obviously big blocker. We'll discuss that in a separate article, so stay tuned! Taking down all the Timeloop Demons might be a problem too, depending on your difficulty setup, so beware. They crop up strewn across the entire region, so you'll need to be fairly thorough in exploring it. You'll know that you've killed all the Timeloop Guardian Demons by the deep rumbling sound that'll appear after doing so.
The good thing about talking it out with Vigil Elder is that he'll give you an item known as the Guardian Detector, which allows you to more easily suss out Timeloop Demons and cut them down. Handy stuff.
Once you're through with all of that, you'll need one of three different Amine Prisms to close down a particular Timeloop:
Upon entering a Timeloop, seek out your companion drone and interact with it to get an anomaly assessment, letting you know precisely what kind of Amine Prism needs to be used to shut it down. Prisms can be found as quest rewards, but also by selecting the right dialogue options when discussing Timeloops with certain NPCs.
Remember, too, to go back to Vigil Elder whenever you shut down a Timeloop to score a bevy of rewards. Do so for each completed Good Deed and Mystery, too. Gildas is nice like that.
Timey-wimey.