A white coat against winter snow is the next best thing to invisibility, and that's important for predators and prey. Arctic foxes are both. They hunt voles and lemmings, and in the high Arctic they trail after polar bears and glean scraps from their kills. Bears, snowy owls and golden eagles eat foxes given the chance, and the foxes' white winter coat helps them to hunt and hide.
Hair is white because it lacks pigment. Animals have cells that produce melanin, the natural pigment that gives hair, skin and eyes color. When the fox is shedding its summer coat and growing its white coat in the fall, melanin production is shut off and the fur comes in without pigment.
Daylight, not cold weather, triggers seasonal shedding and hair growth. Animals register changes in the photoperiod - the hours of daylight - which spurs the secretion of hormones such as prolactin and melatonin. This has been duplicated in indoor experiments - shortening the photoperiod induces hormone production and the growth of the winter coat, and lengthening it simulates the springtime phase.
Come spring, that white coat is a liability, and the Arctic fox switches to a more appropriately colored coat.