I want to attach the same logic from Undertale in a new project, but I have no idea how. Esp. the save and load kind of thing, with all the characters changing some lines based on what I did on other saves. Would appreciate any help :)

This is not a Game Design question, so this is moved to a more appropriate forum.

I don't know the game, but by reading about it, it seems to have several ways to end a level.

To implement you thus need several ways to end a level, sort of different exits thus. To have an effect on game behavior you need to record how the player ended a level, and (obviously) store it. Also, when initializing a new level you access the history, and change some parameters of the next level.

For saving, standard would be in a save file, but then each save has independent history. If you want to share that history data, you could save (or rather, update) a second file with the history, so it becomes independent of saved games.