When Chapter 2 starts, use the book on the bottom right to shadow-transfer to the wall and pick up the wheel from the tree. Interact with the machine on the back left of the room, and attach the wheel from your inventory to it.
The puzzle is divided into three different parts. Here's the answers to all three. Once each one looks the way it's supposed to, press the orange button to test the circuit on the right of it.
* Accidentally selecting a piece of red wire and putting it on the wrong place won't disqualify you from getting the achievement. "Mistakes" only happen when you press the orange button before everything is where it's supposed to be.
** If you make a mistake your first time around, you can revisit Chapter 2 from the Chapter Select screen or by Restarting the Chapter to go at it again