Games have flags (kind of a checklist) of all the possible action you need to do in the game (in the sense of “complete this quest”, “find this object”, “unlock this part of the map”, etc.) And also a bunch of variables to indicate your location in the game, what equipment or item you have in each slot (+inventory).
The save will just be a backup of all those flags and variables and which state they were in at a point in game.
This is a bit of a simple answer and will vary a little depending on the type of game.
Bonus: this is how cheats work too. People will analyse game memory and find where these flags/variables are and offer you to edit the value. Like find the variable “health potion in inventory” so that you can add 99 of them if you like.
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