Pressurisation indeed increases rigidity of the structure. Air pressure expands the fuselage, and applies a tension stress uniformly throughout the fuselage skin.
When in the air, the aircraft is suspended by the wings and garvity attempts to bend the fuselage downwards. The failure mechanism of bending a tube is buckling: the top side has tension stress, the bottom side compression stress, and this is the side that buckles. Internal air pressure helps in two ways:
- The bottom side has pre-tension stress applied by the air pressure. The gravity bending is now resisted by a reduction in tension stress, instead of an increase in compression stress. This results in less deflection difference between upper and lower surface - higher stiffness.
- The fuselage expands as a result of the overpressure, increasing the distance between upper and lower surface and therefore reducing reaction stresses from bending.
We don't want to compress sheet metal, it buckles pretty quickly. However if air presses against it, that helps in preventing that particular failure mode.