How does writing a bit to a GPIO pin turn on my induction motor?
It's hard to say without some type of schematic for the controller board, but what's probably happening on a base level is that the motor is connected to some type of transistor that controls when the fan turns on or off.
A transistor is a semiconductor device that acts like a switch. How much voltage and current you need in order to make the transistor conduct (turn on) depends on the actual transistor. For general use, there are BJT's and MOSFETs. BJT's rely on a set amount of current to turn on, and MOSFETs rely on a set voltage to turn on.
There also exist driver IC's that have something that is called an enable pin. It's just want is sounds like: a pin that enables the IC and allows it to operate. It is possible that the fan is connected to a driver IC with the Rasberry Pi setting the enable pin HIGH when you want to use the motor. These IC's also use transistors to operate, but on a more complicated level since their meant for a specific application.