Intuitively, the hair dryer sucks power out of the wires, causing a voltage drop, causing the lights to dim. But the opposite happens: the lights get brighter. These are LED bulbs. It is the heating element that does this, not the motor (running the motor with no heat has no effect on the bulbs). The lights are on a dimmer circuit.
What causes this effect? I am assuming it has something to do with the LED power regulator or the dimmer, but am curious to know the details.