This is a two step process that must be done during the same playthrough. The first step will be to look at the picture above your mom and inspect the missing puzzle piece.
Much later on in the game, as you crawl through the ventilation system, find the missing puzzle piece in th bottom left of the screen.
See this achievement (and every other one in the video below):8:58
- Click on the picture (top right) and inspect the missing puzzle piece1:17:16
- Puzzle Piece (crawling through the vents; pick up the puzzle piece on the bottom left; only available if interacted with puzzle at beginning)