The steam from the shower's hot water comes into contact with a cool surface(the mirror) and the steam condenses from a gas into a liquid on the mirror.
Explanation:
The fog within the mirror is that the condensation of water vapor because it touches a colder surface. By running cold water you only quiet down the bathtub tub and everything around it. currently the vapor coming back from the recent shower can principally condense right there and can not reach the mirror.