What is the "field-weakening" operation?
... I don't understand what is the final goal.


The final goal is to increase the operating speed of the motor without needing to increase the voltage or current required from the power supply. That is to provide increasing speed with constant power. Since mechanical power is speed multiplied by torque, constant power with increasing speed means reducing torque at increasing speed. 

Most loads require the same torque or more torque to operate at higher speeds. However some of the torque used drive a vehicle is used to accelerate inertia and climb hills. Engine driven vehicles have gear boxes that are shifted to progressively higher gear ratios as the speed increases. That is constant power operation. At higher speeds, the rate of acceleration and the ability to climb hills is limited. With electrically driven vehicles, the use of gear shifting can be avoided if the motor can be operated at constant power in the upper part of the speed range. It might be possible to operate over the same speed range without sacrificing high-speed torque, but that would require expensive power capability that only be used a small percentage of the operating time.

Another use for a constant power drive is driving the center of a spool that is accumulating paper or some other material with a constant surface speed and tension. Since power is surface speed multiplied by tension, the required power remains constant with speed change. An empty spool has a small radius, so the shaft speed is high and the torque is low. As the material accumulates, the radius increases and the shaft speed is reduced to maintain constant surface speed.