The second room of the Nursery contains a flytrap, which can be pushed left and right by Klaymen, and five rings hanging from the ceiling (which are food for the flytrap, as Willie states in his letter). The goal is to open the door leading out of the Nursery.
The door is not opened with the button on it, but by pulling the fourth ring (second from the right). In order to keep the ring down, Klaymen must push the flytrap under it and then pull the ring. The flytrap will catch the ring and keep the door open.