Hell Is Us is, shockingly, one of the few games that actually successfully leans into its puzzle-heavy gameplay loop. You get no ready-made maps on-the-go, no tips and tricks on how to go about solving problems, and no quest logs to reference whenever you want. Instead, all you get are inferences and suggestions, and a whole bunch of ways to make heads and tails of them.
Crucially, sometimes you'll simply have to complete a particular puzzle to progress in Hell Is Us, as is the case with the big Watcher's Nest puzzle. This ties directly into the game's grander progression scheme, with a way to permanently (!) stop enemies from respawning in a particular region. A heck of a feature, but you wouldn't know it! We're here to help, though, with a reasonably spoiler-free explanation of how to solve this puzzle.
The Watcher's Nest is basically the crux of Remi's progress up until this point in the game. A whole host of leads and developments tie into this location, and we highly suggest exhausting all the dialogue options with all the NPCs you come across to figure out what's actually going on here.
Obviously, you can (and maybe should?) figure out the puzzle at your own pace to make the most of it. Yet, this one definitely feels a tad overwhelming at first, and we understand why you might want to skip the figuring out portion of it. So, without going into the nitty-gritty of it, the Watcher's Nest Puzzle in Vyssa Hills is all about slotting four unique ceramic flowers into the right slots on the hideout's walls.
First, you need to explore the entirety of Watcher's Nest and go down a ladder deeper into the crypt. There, you'll find a mysterious lever that won't do anything at that point, and then search the other surrounding sarcophagi for items. One of them will hold 4x Ceramic Flower items, and these are the bobs you need to fiddle around with.
Head on back to the entrance to Watcher's Nest from Vyssa Hills, and let's begin:

The first Ceramic Flower goes into the third crest from the left, with the Grief Symbol (as illustrated above). Look for it in the entrance area, right around the fancy round stone table.

The second Ceramic Flower isn't far away from the first one. Head into the hallway to the right from the round table and look to your left, as shown in the picture above. Slot the Ceramic Flower underneath the shield engraved with a book.

The third Ceramic Flower needs to go deeper into the hideout: look for a wall with a fancy blue banner hanging about. It's going to be flanked by two flower inserts, but you want to slot the flower into the one to the right, relative to the banner itself.

Finally, the fourth Ceramic Flower needs to be inserted into a cheekily hidden spot, just behind the big whiteboard used by the deceased members of the Vigil. It can be found in the central chamber of the hideout, in the room with all the computers and assorted material strewn about. Look behind the whiteboard and then insert the final Flower into the left slot, the one with the plaque reading 'Algea' next to it.
With that all sorted out, there's going to be rumbling noise inviting you back down the ladder, into the crypt where you initially found all those Ceramic Flowers. Interact with the big lever to open the central sarcophagus, collect all the items inside, and then head back up. As you ascend, you'll hear the sound of a toilet being flushed, and then meet a new character in the Watcher's Nest proper.
This one’s important.