"Handling speeds" relates to the speed range airplanes can use when maneuvering around the airport to land on a given runway in the circling configuration (usually gear down and partial flap), as given in the 4 categories, and "circling area development" relates to the radius of the turning circle that results when maneuvering at a given max speed.
The max knots for each category relates to the ability to circle and keep the airport environment in sight and maintain obstacle clearance at that speed, allowing for the radius of turn at a standard turn rate at the maximum speed for the category.
The Category you use is based on the slowest speed you are able to use in the landing configuration at Max Landing Weight, and you use the category whose max speed is at or more than your max gross weight's reference speed, if a reference speed for that airplane is published (if no reference speeds are provided, you have to use 1.3 VSO at max landing weight). You want to use the slowest category you can fit into, since that gives the lowest minima.
For example, in the CRJ, the Vref at max landing weight ranges from 120-141 kt, from the airplane's Speed Book. So based on that you could use Cat C minima right up to the Max Landing Weight of 47000 lb where you just squeezed in (In the CRJ you actually circle at 160kt, the minimum speed for Flaps 30, and you go to landing flap as you got established on a base leg for the target runway and ready to commit, and slow to Vref plus a little margin extra for turning allowance on the turn to final).