This all has to do with the amount of energy the water contains. The easiest way to add energy to water is to heat it (thermal energy). The hotter the water is, the more the individual water molecules move around and collide with each other. When a dissolvable solid like a bath bomb is placed in water, the water molecules that surround the solid attract the exposed individual molecules of the solid and pull them away from the bulk material, thus surrounding and dissolving the solid particles. Hotter water does this faster than colder water because the water molecules have more energy and are moving around and colliding faster, thus they interact with the solid particles quicker and more effectively.
A bath bomb is basically composed of two major ingredients, a weak acid and a bicarbonate salt. Both of these readily dissolve in warm water and react to form water and carbon dioxide:
R-CO2H + NaHCO3 --> R-CO2Na + H2O + CO2
The CO2 bubbles away and gives the bath bomb the fizzing action. Other oils and scented chemicals are usually added for fragrance.