The striped thing between the motor and the dc generator is a shaft which couples them together mechanically. The motor drives the generator (tachometer) via this shaft.

The set speed control provides a dc voltage, say 12 volts for maximum speed and zero for stationary. This could be a potentiometer providing any voltage in a range from zero to +12 volts.

The difference amplifier will amplify any difference between its two input voltages.

If the motor is stationary and the speed control is moved from zero to half speed then, since the tachometer is not rotating and not producing an output voltage,  there will a difference in voltages at the two inputs of the difference amplifier.
Therefore there will be an output voltage from the amplifier.

Since this voltage is not high enough in value to drive the motor, it is increased in amplitude by the dc amplifier.
A dc amplifier is a special type of amplifier which can increase dc voltages.
For example, 10 volts dc in could give 100 volts dc out.

This amplified dc powers the dc motor which begins to increase its speed of rotation.
This in turn rotates the tacho which produces a voltage proportional to speed.

As the tacho voltage increases it will eventually reach the same value as the "set speed" voltage.
At this point there will be no output from the difference amplifier and dc amplifier. The motor is up to the correct speed.

However, since the motor is no longer powered by the dc amplifier its speed will start to fall.
But the tacho output voltage will start to fall, and there will again be a difference between the two input voltages to the difference amplifier.
This will produce an output from the difference amplifier and dc amplifier which will power the motor and correct this drop in speed.

In a poorly designed system this drop in speed and its correction can cause "hunting", a regular variation in speed.