The Navigator is the brains of the operation and controls everything. Reading about the functions here is a good place to start.
The ESC takes DC voltage from the battery of the ROV and a PWM throttle signal from the Navigator and converts it into three AC phases to drive the brushless motor.
Not shown in that diagram are the two additional ESCs for the vertical thrusters #7 and #8 for the Heavy configuration. Those are connected to output channels #7 and #8 on the Navigator.