I want to learn the function of the gate driver here in this circuit.

P or N MOSFET transistors or those in the newer generation, IGBTs, have a large input capacitance between gate and source. These transistors are driven by voltage, therefore -opposite to bijonction transistors which are driven by the base current. In fact, until the input capacitance is "filled-in", the MOSFET will not saturate (this is the scope of driving the gate to up to 15V). This is why, there are necessary drivers that can provide large currents (1A or more). If the C_IN = 1nF, to have 15V on that capacitor in fifteen nanoseconds you will need 1A. Why 15V? at this voltage is produced deep saturation, with a minimum V_DS 1V, 2V or even sub volt.