How do I solve the Puzzle?

Solution 1:

You must collect every little purple block (as Berg mentioned in their answer) in chapters 1 - 16. I detailed where every piece is in this answer. When you collect them all, you must align them in the "Puzzle". When a piece is in the correct space, it will start glowing. The final result is:

IWASBORNINTHEUSA
This will end up unlocking different dialog (a secret ending) at the end of Chapter 19, if you examine the computer before taking the ladder down.

Solution 2:

There are little glowing collectables in the game (picked up with right mouse click). (I found them while using the Rasmus mask [2nd from top]; I don't know if you have to have it on to see them.) There is one in Chapter 6, second floor, on the balcony just after you climb out the window. It is under the second tree, but should be clearly visible. It will be a letter.

Each of these letters will slot into the puzzle accessed by pressing Escape in game.