One site explains it this way:
Long before mirrors were invented, people used to see their reflections in pools of water. Since they had no scientific knowledge at the time, they believed this reflection to be their soul or their other self. And they believed that any interference with the other self would injure it. Hence when mirrors were invented, and they were broken, it was thought that the other self was harmed. And where did this seven year thing come in? Well, the Roman belief was that life renewed itself every seven years. And, since a mirror meant �broken� health, it was believed that the person who broke it would need seven years to recover....