That's some lovely artwork, nice one!
A very simple checkpoint implementation would be to have a Node2D as a respawn point, then when the player enters an Area2D, you can heve the respawn point move to the player's position. When the player dies, reload the map, but keep the respawn point where it is, and move the player there once the map's reloaded (or you can not reload the map, and have the gamestate stay the same).
There are lots of little tweaks you could add on, like storing whether certain buttons should spawn pressed or not depending on the checkpoint reached.
I'm not sure how clear that is, but I hope it helps!