A simplified explanation to your questions (not considering actual energy losses, efficiency, etc.):

The thrust or pulling force of any propeller comes:

1) From the area of the rotating disk times the differential pressure of that stream of air before and after the prop.

2) Or from what is the same: the mass of air moved by the disk times the differential velocity of that stream of air before and after the prop.

The power that the propeller receives from the engine is rpm times torque (twisting force at the shaft), and that is constant for each throttle position.

Max torque occurs at rpm lower than max rpm.
Max power occurs at rpm higher than the rpm’s of max torque.

An engine should be loaded with a propeller able to move as much air as it is possible to be moved for that max torque or max power.
That is achieved by balancing the jigsaw formed by the area of the rotating disk (which is proportional to the square of the diameter of the propeller) versus differential pressure (which is proportional to the rpm’s and to the angle of the blade (pitch of the propeller)).

Big diameter propellers of low pitch: Better efficiency, more “air grab” at low forward speed of the model, higher breaking effect during landing, better engine idle, good for 3-D and high drag models (it is like the first gear of a car’s transmission).

Smaller diameter propellers of high pitch: Lower efficiency, less “air grab” at low forward speed of the model, lower breaking effect during landing, poor engine idle, good racers and low drag models (it is like the highest gear of a car’s transmission).

That is the reason for which the engine manufacturer gives us a range of propellers that will perform good.
For diameters above that range, the rpm’s will drop below the max torque.
For diameters below that range, the rpm’s will go too high above the max power, with possible damage of the moving parts.