Explanation: When you “pick your nose” you stick your finger in your nose and, well, remove whatever is in there. “To be picked on” means that someone is bothering you, bullying you: The older kid picked on the younger kid by pushing him every time they saw each other.
This joke plays with the words “pick” and “nose” and the double understanding there. It is funny because the nose is being picked on, not being picked.
It’s more polite to use a Kleenex.