Because your hydrostatic load changed
The RPM changed instantly because the hydrostatic load on your prop changed. Because air is a fluid. Effectively, your prop is a fluid coupling.
An example of "not a fluid" is an engine geared to cogs on a rack railway. An RPM change must match a speed change. Not here.
Compare to your car's automatic transmission torque converter- the most common fluid coupling in normal life. It intentionally has "slip" so the engine can idle at 600 rpm while the car is stopped. This slip varies by engine force: if you push the accelerator, the engine will rev more while still stopped. On a slight downhill, it slips less, and on a steep downhill, it can actually push (windmill??) the engine*.
There is a relationship between throttle position, engine RPM, and hydrostatic load (resistance).
Your propeller is the engine side of a torque converter.
When you pitch down, you change induced drag, i.e. the airflow is resisting less. You have instantly removed hydrostatic load from the prop, so the engine RPM instantly increases.
Put it another way, at any given altitude, weight, pitch and throttle position, the airplane will seek to "balance out" at a particular RPM and airspeed. But it won't get there instantly. You can change pitch angle very quickly, but it takes awhile for airspeed to change.
So in that moment when you have pitched down and you have not gained airspeed, the air is resisting less (because the airplane wants to go downward on its own, like going down a hill on a bike)... so the RPM must increase.
Again, air is not a rack railway.
* however you can't "windmill" start an automatic transmission car. That's because the gears are engaged hydraulically by a pump on the engine side...