Explanation: The Easter Bunny is a character from folklore in the United States.  He is said to bring eggs to children on Easter Sunday.  Soggy means wet.  So how would a wet rabbit dry itself?  A hair dryer, perhaps.  Or in this case, a hare dryer.

Hare and hair sound the same.  Hare is an animal that looks similar to a rabbit; hair grows on your head.

This joke is funny because hair and hare sound the same.  Happy Easter!