The "paper" in this context is a quest item required to solve the room-swapping puzzle in one of the game's more complex locations.
For a deep dive into the story details contained in these papers, you can check community discussions on Steam. The Vanishing of Ethan Carter Redux
: The objective is to select the correct room configuration to restore the house's original layout. If the correct choice is made, the portal disappears and that room is "fixed". The "paper" in this context is a quest
: Many players recommend using real-world grid paper to map out the interior and track the various room permutations. Other "Paper" References in the Game The Vanishing of Ethan Carter Redux
: You must find and place down a piece of paper to activate the portal mechanism .