Why does RPM for a fixed-pitch propeller change with an aircraft's pitch?
The driving variable here is airspeed, not pitch attitude per se.  Your airplane naturally goes faster when you put the stick forward and decrease the wing's angle-of-attack, and this changes the prop rpm.

Your pitch input is changing the wing's angle-of-attack which leads to an airspeed change.  Even with the engine switched off, the aircraft would fly (glide) at a higher airspeed with the stick forward than with the stick aft.  And for a given throttle position, a fixed-pitch prop turns faster at a high airspeed than at a low airspeed.  

Just as a wind turbine does.

Your tachometer can be used as an airspeed indicator.

The key factor is the speed of the air flowing through the propeller disk-- with a fixed-pitch prop, this dwarfs any effect related to the amount of air being forced into the engine.

And yes, this effect can be explained in terms of the angle-of-attack of the propeller blades.  Increasing the airspeed decreases the angle-of-attack of the propeller blades, "unloading" them, or in extreme cases subjecting them to negative loading so that the prop is driving the motor rather than vice versa.

In a fast dive at a low power setting, the wind (airflow) is making the prop move, not the other way around, and you'd pick up more airspeed if the prop fell off.  But even at a lower airspeed and higher power setting, the speed of the oncoming airflow has big influence on the speed of the prop's rotation.