Mica’s iridescence is a result of thin-film interference, defined as a “natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, either enhancing or reducing the reflected light.” 

If the waves reflected off both parts of the are in sync, they will amplify eachother, enhancing the reflected light (constructive interference). If they are opposite each other, they will cancel eachother out, reducing the reflected light (destructive interference).
When the white light which consists of different wave lengths (colors) hits the surface, some colors are intensified while others are attenuated. Seeing the surface from a different angle changes the relationship between these waves, which determines which colors we see and which ones get blocked out. (Wikipedia).