The developer retroactively adds the achievements which they can configure around things like save file determinations. This may house back-end stats and the achievements are built around the data they can record. So if a save file contains information about the user getting in excess of 1,000 kills or completing the game on hard then they can create an achievement which detects such.

It's how the devs set it up. As long devs put check system in their game they can use that same check system to toggle steam API, that how Steam knows.

It's possible for a game to check to see which achievements you have been awarded for that game. Somewhere on Steam is a game that uses achievements as a checkpoint save game system. But it's mostly used so the game can verify that achievements you've earned in the game were properly awarded on Steam, and if not, try to award them again. So if the recently patched game is doing this, it will know exactly which ones need to be updated with Steam.

Of course, the game could be lazy and just update all earned achievements every time you launch the game. Anything already showing on Steam shouldn't pop a notification so you'd only see new ones if there were any.