As A tech, I like to explain to the client why we do things the way we do. A picture is worth a thousand words:
Tires are designed to carry a specific weight with a specific air pressure.
Too little pressure and the tire flattens out and the sidewalls flex and your handling goes wonkers. Off-roaders, when driving over soft sand, let air out of the tires to give a larger patch so as to not sink in.
Too much pressure and the tire does flatten and your contact patch with the road is smaller. A smaller patch means less traction. On the other hand, a smaller patch and stiffer sidewalls mean less tire flex, less heat buildup, and less rolling resistance. For highway use you increase tire pressure for increased gas mileage.
The compromise, in the middle, gives you a larger contact patch for the most traction while maintaining a reasonably stiff sidewall for precise handling. With the weight of a passenger pushing down on the rear tire, you need a few more pounds of pressure for the tire to maintain the optimum shape.