Koyovis explained it with maths and cool diagrams, here is my layman's version, if it helps.
You're confusing aircraft drag with the drag on the prop. Since the prop is powered, it typically produces thrust not drag with respect to the aircraft. It only produces a net drag if it's windmilling (that is being turned more by the airflow than by the engine) - which is why you would feather the prop on a dead engine in a multi-engine aircraft.
Each prop-blade however produces drag in the direction they're travelling (i.e. sideways) in order to produce lift in the direction the aircraft is moving (i.e. thrust) - and it's this drag the engine is fighting. Increasing the pitch makes it harder to swat that thing through the air (but also produces more thrust) thus the engine RPM goes down.