Not sure what you are looking for in an explanation but your first picture is of a fuse holder. The fuse is in line with the variable output voltage circuitry and hopefully will protect the whole thing from catastrophic failures like shorts or overloads.
The second picture is of the printed circuit board that contains the components that are used to make the output voltage variable. Components like resistors, diodes, capacitors and a Diac.
The third picture is of a Triac or Alternistor. This is the solid state power device that actually switches the AC voltage on / off every half cycle of the 60 Hz power. To reduce the voltage to the load (router, light bulb, soldering iron or what ever) the circuit delays the turn on of the AC source to the load every half cycle by variable delays determined by the position of the dial on the variable resistor called a potentiometer. This whole method of controlling the voltage to the load is called “Phase Control”.
The third picture is of some of the components mounted on the Printed circuit board that are used to control when the Triac or Alternister is turned on in each half cycle.
By reducing the voltage to a router, any universal motor, you cause the magnetics in the motor to “slip” which in turn reduces the power available which reduces the torque and horsepower which reduces the speed.
I tried not to get too technical and but still explain how these “speed controls” work.