As children, most of us had parents that would make us wash our hands after almost everything we did. As adults, we know that they were not joking around, because hand hygiene is of more importance than any child could know. Without proper hand hygiene, germs and bacteria can stay on your hands and be transferred to the food you eat, the itches you scratch, and almost anything else you touch.
Cleanliness is only one factor in maintaining good hand hygiene. Of course you want clean hands to not get other things dirty. However, beyond the need for clean is the need for sanitary. If you use a public restroom for example, you don't know what the last person who flushed the toilet has for hygiene habits. The only way to protect yourself from other people's bad habits is to stick to your own good ones.