It's a little different for steam/xbox/ps, but generally there's two kinds of achievements. Trigger achievements and value achievements.

The trigger type ones are the ones that fire once, like when you beat a level or get a certain item or something. These basically have an api call like TriggerAcheivement(achievementID).

The value type ones are the ones like collect 1000 gold or kill 500 enemies. You could choose to handle this manually, and then just fire an achievement trigger, but this usually becomes complicated with multiple saves or crossplay, etc. So instead, there's an api call like IncreaseStat(statID, 1). And you call that when you kill an enemy or whatever and once it's been called 500 times, the achievement activates.

You usually set all the achievements/stats up in the steam/xbox/ps backend and then your game is just responsible for calling those api functions when appropriate.

I’ve always been curious about this myself.

Pretty much