What is the function of a MOSFET driver in an H-bridge?
An H-Bridge contains 4 transistors. Often these are FETs. The gate of a FET is not as simple to drive as you might think, for two reasons:


FET gates have some capacitance, which takes a little while to charge up to the threshold voltage. This in turn means the FET take a little to switch on. While the gate capacitor is charging up, the FET is only partially on, and therefore dissipating power and getting hot. The faster you can charge that capacitor, the faster you can switch it on, and the less power you waste. You'll see that FET drivers, like the TC4426 advertise that they can drive very high currents into the gates.
The high side FET on an H-Bridge needs a high voltage to switch it off. This is almost always too high for a normal logic circuit to interface to. The driver may also act as a voltage level converter.