In this case, we say "dirty" because there is dirt on their hands. If we say messy, it means "not the way it should be " = clean. In the every day life, we just say dirty.

Dirt is filth, bad stuff which we hate: mud, dust, soil, poo, goo from behind the refrigerator, it is usually full of bacteria. Mess is stuff which is in the wrong place: when I wake up in the morning, my hair is messy (but never dirty!). Children's hands can be dirty after they have been playing outside, or messy after mealtimes.