My eyes usually go watery when I'm at a supermarket. Why is that?

Dry eyes. Large buildings can be drafty and have air conditioning - this dries out the eye and causes reactive tearing (epiphora). Seems paradoxical but the large tear gland is designed to be activated automatically with irritation.
Dry or inflamed eyes. Dry eyes can become more irritated in bright lights. Additionally, people with certain inflammatory conditions of the eyes can feel worse when the strong fluorescent lights affect them. If the watering does not go away quickly or with artificial tears, you should have your eyes examined.