Many proofs are known.
This one, nearly without words, as a dissection. The proof is done by just translating the different pieces.
Just to prove they exactly fit :
From O, midpoint of the square on the large side, draw the parallel MN and the perpendicular to hypothenuse.
MN=AB and pieces 1 and 2 fit after translation.
AM=BN, hence AC=AM-CM=BN-CM is really the side of the "hole" in the puzzle.
Another visual proof, even more simple as no formal proof is necessary : the pieces fit "automatically".
The empty part is a² + b² to the left, and c² to the right.