Other components can also be used but MOSFETs are close to being perfect switches, when off they conduct very little and when on they only drop a tiny voltage. Consider a switch passing 75A, if it's resistance was only 0.1 Ohms it would still drop 7.5V and lose over 500W of energy as heat. The resistance of a fully conducting MOSFET is very much smaller than that so less heat and more efficiency is possible.
Other components usable as power switches in inverters are bipolar transistors, SCRs and IGBTs.
To make a MOSFET work as a switch you have to control the voltage between the gate and source pins. Exactly how you do this depends entirely on the remaining circuit configuration.