The solubility of air in water depends on temperature. When water that is saturated with air is heated, the air will bubble out of the water as the solution becomes supersaturated. We’ve all seen this watching a pot of water heating on the stove. Long before it boils, bubbles of air start forming on the bottom and sides of the pot.

In your shower, hot and cold water start out saturated, but when they mix, the warm water can be supersaturated, causing tiny bubbles that make the water look milky.