I really wish people would quit making the analogy between a CS prop and a transmission, it's nothing of the sort.
Look, these are propellers, you are pilots, you should understand them correctly.
A propeller is a screw, the steeper the pitch, the further the screw will try to pull through the medium it is in per revolution. Horsepower is a measurement of how fast you can move something in a given time. The further you moved it in a given time, the more horsepower it required.
Propellers are an airfoil, not a gear. When you increase Angle of Attack, you increase drag, when you decrease angle of attack, you decrease drag; simultaneously you coarsen or fine the pitch of the screw.
This is the basics of a Variable Pitch Prop. When you use a Constant Speed Propeller, you use this ability to vary the drag profile in order to govern the speed of the engine by using up all of the available torque.
The propellor blades are mounter so they can rotate. In the middle of the hub is a piston that drives out with oil pressure and returns either through a spring in the hub, nitrogen charge, or some counterweights on the blades. There is also a linkage between the piston and the blade to adjust the pitch. That prop hub is bolted onto the end of a crankshaft with a hollow nose and an oil port through the front main bearing to allow pressurized oil to be fed into the hub, or released from the hub.
To control this oil pressure and thereby the pitch of the blades and drag of the propellor, we employ a variation of James Watt's centrifugal governor. This consists of a set of flyweights counterpoised with a spring whose rate can be adjusted. These drive a spool valve which controls the flow of oil in and out of the hub. The flyweight shaft is driven off the crankshaft.
What you do when you move the blue handle is adjust the tension of the counterpois spring. As the engine speeds up, the flyweights want to go out and open the valve to put oil into the hub and add drag. When rpm drops, the counterpois spring brings the flyweights back in again releasing oil from the hub and reducing blade drag.
When you move the blue handle and adjust the counterpois spring, you adjust what RPM the spool valve opens and closes at. Here is the really important part to understand: When you select a lower RPM, you are selecting to increase load on the engine, that is why you see a rise in MP when you pull back the prop.
You can get yourself into trouble with detonation if you are making too much power when you pull back. Pulling back and reducing RPM also changes the timing of the flame front vs Top Dead Center which will also move you closer to detonation.
This is managed by running high RPM when in high power modes, and reducing torque (Internal Cylinder Pressure) to operate in low RPM modes. You can reduce torque one of two ways, restrict airflow with the throttle or restrict fuel flow with the mixture, I prefer to use the mixture.
Also,
since Power=Torque*Time, an engine turning more RPM will be able to make more power, so when climbing is necessary, I increase RPM rather than increasing torque with mixture or throttle.