This question has been intriguing me for some time. In Autogyro the rotary blades merely acts as what fixed wings act in normal aircrafts to generate lift. So why not simply have a fixed wing instead of rotary "wings"? What advantage do this rotary nature of wings provide compared to fixed wings? In both ordinary planes and autogyros the propulsion is generated by the front or aft engine.
Autogyro blades have a higher airspeed than the aircraft that is carrying them, so they can generate lift even when the aircraft speed is low. They fly as if they had a wing the size of the whole disk, not just the blade area - yet can be stored or transported easily.
They can land (but not take off) almost vertically, which is a useful safety feature in the event of an engine failure.
Having said that, they aren't common because they don't do anything useful significantly better than the other options, except perhaps being 'different and interesting'. Flying slowly isn't terribly useful.