You don't need this shield to drive servo motors, they have their own controller, you need just a 5V power line, GND and digital output pin to generate a PWM and that's it. Usually Arduino has enough power to drive servo motors.
This shield you are talking about is used for DC (direct current) and stepper motor. These motors need power gain and the shield provides it and often has a chip called L298N (which implements H-bridge) with MOSFETs to properly feed the motor.
What does this shield really offer in the way of driving motors?
A proper circuit and components so you plug your motor and connect into your board leaving you just the programming task to control motor's position and speed.
What is lost by not using this shield?
Actually, you use this kind of component just for practice purposes. It is already built and you just use it. However, it's a good learning process building the H-bridge yourself and understanding how it works and why it is necessary (if you have time, this is a good task).
What if I my project has more than 2 motors?!?
Well, just decide how many H-bridges you need. This shield has the capacity for 2 DC motors or 1 stepper motor. In case they are servo motor, you just use this lib. It is possible to have plenty of servos working together and all you need are good battery and enough digital pins. They share timers of microcontroller, that's why it works for many motors.