Hot water helps get grease off dishes and emulsifies it to help it from sticking to other dishes in the water. It also aids in softening food left on the dishes.
Another way to get grease (and bacteria) off your dishes is to add salt to the water. Salt is a natural antimicrobial, which is why it is used in canning processes (hence why canned food is so salty).
It's more important to have warm/hot water for the rinse since it will help the dishes dry faster keeping them from possibly getting mold on them.
Of course, you could always get a dog to "pre-rinse" the food off your plate before washing them