Why is it better to use this Opto-coupler to drive the gate of a FET than using other methods?
An opto-coupler offers common ground isolation between sections with different voltages and currents. Also isolation between the digital and analog sections of a board, allowing digital control of analog circuits, and feedback of analog levels to a digital side with a MPU in control of analog circuits.

They also offer isolation from the high and low sections of an H-bridge, as the upper MOSFET is not at common ground, but usually between Vcc and ground, rapidly changing with the pulses. As shown in the drawing the MOSFET gate driver is powered by an isolated supply, so the gate can be properly driven regardless of the voltage at the source pin. Some half and full bridge driver IC's offer an internal offset and raised voltage for the upper MOSFETs and IGBTs, but they have voltage limits and may not be suited for a supply rail of +2,000 volts.

Optocouplers solve many connection problems where common grounds are not the same, or cannot be shared except at the power connector where all the various power grounds merge.

The optocoupler you refer to is designed to drive medium power MOSFETs with up to so many amps of gate drive current. It is not a general purpose optocoupler where a sharp rise and fall time at high current levels are not needed.

Other methods work but have a higher parts count. A transformer coupled design can be found with dual secondary windings, but additional Schottky diodes and resistors are needed so the MOSFET gates see only a positive pulse with a fast ON time and faster OFF time. This design is more common if IGBTs are used at very high voltage levels, far beyond what IC's can tolerate.

Extra transistors used to level shift the voltages for the upper MOSFET get complex, so if the voltage is not too high IC's that do that can be found for low to medium power applications.