I looked up 'put together' in the Merriam-Webster dictionary.
1 : to create as a unified whole : CONSTRUCT
2 : ADD, COMBINE
If so, does 'put together' mean #1 in 'put a puzzle together' and #2 in 'put puzzle pieces together'?
They put the Lego castle together.
put together is a phrasal verb here that means to assemble.
They assembled the pieces of the puzzle. They put the pieces of the puzzle together. They put together the pieces of the puzzle.
pieces of the puzzle just sounds better than puzzle pieces although that too can be used. Obviously, pieces here would be blocks or shapes made of wood or plastic. They can, of course, also refer to jigsaw puzzle pieces.