It happens when the "silver" on the back oxidises and its called foxing. There isn't really any way of stopping it, but it wont spread over areas of more than a couple millimeters. Most people expect this when buying an antique mirror because it shows that the mirror still has its original glass. It defiantly doesn't detract from the value of the mirror and can look quite attractive. No need to worry.

Look at my source below, it contains creative ways to conceal the black spots on the mirror. You cannot fix them, but you can creatively cover them up.