Bath bombs are hard-packed mixtures of dry ingredients which effervesce when wet. They are used to add essential oils, scent, bubbles or color to bathwater.
The part that makes them fizz is the same as an Alka Seltzer or soluble aspirin tablet: sodium bicarbonate and citric acid. In water, these chemicals dissolve and react with each other to form sodium citrate and carbon dioxide. The sodium citrate stays in solution and you don't really notice it, but the carbon dioxide bubbles out as a gas that helps the bath bomb break up. This lets the detergents, perfumes and oils that make up the rest of the bath bomb mix with the bathwater.