Why must the throttle be moved slowly?
There are a few ways rough throttle handling on a propeller airplane can get you into trouble:


Gyroscopic yaw and torque: a rotating prop creates yaw 90 degrees to its spin which are offset by control inputs (mainly rudder), throttling up or down requires changes to these control inputs. Quick changes require bigger changes, slower changes give a pilot more time and allows smoother transitions. This isn't that big a consideration on lower-powered aircraft but it is on high-performance aircraft. In the first case that comes to mind an inexperienced P-51 pilot rammed full throttle in on a go-around and it flipped the aircraft on its back. At low airspeed it's possible there wasn't enough control authority to manage the forces even if he had been prepared for them
Pitch changes: most aircraft will have a pitch change when the throttle is increased or decreased, again quick changes require coarser changes in controls while slower changes mean smoother and better controlled inputs. Again more applicable to higher performance aircraft
Engine and carburetor: The engines in most light propeller-driven airplanes represent the pinnacle of 1950's technology. Fuel mixture, priming, and throttle control are all completely manual - there are no electronic systems to help smooth things out during throttle changes. Many airplane engine carburetors do not have an accelerator pump to shoot extra fuel to prevent the engine stalling during a quick throttle up. Given that an engine stall when airborne is a life-threatening emergency it's in your interest to make smooth, controlled changes


Although not usually a safety issue wear and tear is also a consideration. Smooth, gentle changes are better for the levers, plungers, and cables in the engine control system. Less wear and tear means better reliability and less costs.