Fan control circuit with Arduino
This is an electronics question, and it should be forwarded to the electronics stackexchange.

The fan needs to have a proper amount of current to spin, and because the microcontroller that controls the fan draw very little current on its output pins (check the datasheet for the exact value, but it should be a few milliamps), unless you have a miniature fan, wiring it directly on the microcontroller will not be able to move the fan.

And this is by design that microcontrollers cannot provide much current. Because to provide larger amount of current than what they do, they'd need to have bigger copper tracks between the transistor to avoid having they internal circuits building too much heat, and they would be much much bigger.

So as a work around, we use an transistor, which receives on the "Base" pin a voltage, drawing almost no current, which will "connect" the collector and emitter pins where you can have a much bigger current flowing (though be careful to respect the datasheet's nominal values for current! Or use either a MOSFET, a Triac or a Relay for bigger currents).