Simply speaking, with the puzzle pieces on the table the individual overturns the pieces and the brain draws conclusions for you to form the basis on which you would separate and group the puzzle pieces. It could be based on colour, design, or shape of the puzzle pieces. For example, shades of blue on one pile and shades of yellow on other.
Technically speaking when you unbox a Jigsaw puzzle and watch those small colour pieces lying motionless in front of your eyes, your brain immediately kicks in the processing of scanning. It begins to recognise the pieces that are overturned, pieces that have same and different colours, pieces that have different patterns, pieces that have different shapes and more. This passive activity engages the frontal lobe of your brain. The prefrontal cortex located in the frontal lobe is responsible for performing higher-level cognitive functions. The motor cortex is also located in the frontal lobe is involved in planning and coordinating movement. Along with the scanning the puzzler is moving both his hand in order to move pieces around. This increases the dexterity which subsequently fuels neurons in both the brain hemisphere.