Cells have mechanism to repair damaged DNA. So why don't the cells in moles (the spots), repair themselves and the moles dissappear?
DNA repair is a collection of processes by which a cell identifies and corrects damage to the DNA molecules that encode its genome. However it is yet to be linked closely to mammalian moles.
Information here, states:
"Moles are overgrowths of skin cells called melanocytes, but the genetic factors involved in their development are not well understood."
The BRAF gene is one of the most well-studied genes involved with benign moles, if a mutation arises in this gene - there may be a production of an altered protein that causes melanocytes (melanin-producing cells located in the bottom layer of the skin's epidermis) to aggregate into moles.
I cannot answer why DNA repair does not occur. Maybe the cell cannot identify the mutation, maybe the cell cannot corrects the damage, or maybe it is also from environmental factors. This is an active area of research, but receives far less attention compared to cancer or heart disease.