When your brain has assembled the data through visual scanning, it begins to integrate the information so as to figure out placements. The brain begins to recognise the placement and relays the information to motor neurons so as to make the movement to pick the piece and place it at the identified place.
Technically speaking the parietal lobe integrates visual and motor information which translates in to moving the pieces for assembling them. Jigsaw puzzle helps fuel neurons in the parietal lobe so as to help in aligning and joining the pieces to construct into a unified image/structure.
Parietal lobe is located right behind the frontal lobe and is involved in processing sensory information and in movement. In the initial few steps of solving a Jigsaw puzzle, the puzzler is building deeper and stronger neural pathways between frontal and parietal lobes of the brain.