Two regions that matter for switching
In the active region the collector current follows Ic = beta x Ib, where beta, also called hFE, is often between about 50 and 300. In saturation the collector current is limited by the load instead: Ic = (Vsupply minus about 0.2 volts) / Rload.
Designers pick a base current comfortably above Ic(sat) / beta so the transistor is firmly saturated even if beta is lower than expected. A saturated switch drops very little voltage, so it wastes little power as heat.
Try it: set the base current to 2 milliamps and move beta from 50 to 300. At low beta the fan runs partly on; once beta x Ib passes about 295 milliamps the saturation light turns on and the current stops rising.
Simplifications
This teaching model treats the fan as a fixed 40 ohm resistor and assumes a constant 0.2 volt drop when saturated. Real motors draw extra current when starting and need a flyback diode to protect the transistor when switched off.