If there was no engine, or you were considering a wind generator/windmill type thing, it would be intuitive that it will windmill faster, the faster the airflow, would it not? All the engine does is add (somewhat) fixed amount of power.
Alternatively, consider that the pitch of the prop gives some set amount of forward motion per turn. In order to go faster, it must turn faster.
If you slow the aircraft down (e.g. by pointing it straight up), the angle of attack of the blades will increase, creating as you say, more drag - that's drag in the sense of resistance of the prop to turning. Drag in the plane of the prop disc; hence the engine slows.