The first step of the solution is to complete the Paint By Numbers grid. It's a little tricky at the start, but then everything falls into place pretty quickly. You end up with a completed grid not unlike this:
Puzzlers will have noticed by now that the grid is 66x77, a rather odd set of dimensions for a PBN, and that the first 6 rows are completely empty. It also looks like there are squared-off edges in the completed grid. This, combined with the specific "cut out for" and "reconstruct", should clue solvers that they're dealing with a jigsaw puzzle in the completed grid. There are 42 11x11 squares.
To make it easier for solvers, I constrained the pieces to be in the correct orientation; i.e., everything is right side up. Rearranging the pieces to resolve the image gets you this:
The solution to the puzzle is SQUIRREL.
Once the puzzle has been trained, the answer transforms to SQUIRM.