Why do nurses have to be clean?

Nurses have to be clean because it reduces the risk of transmission of any pathogens to other patients. By frequently washing their hands it reduces the amount of bacteria on it that could be potentially harmful to susceptible patients.

This keeps not only the nurses safe from anything their patients' have, but also the patients themselves. Since usually the only people in the hospital have serious medical issues they need their bodies to focus on their recovery and not dealing with a potential infection that could result from a pathogen.