How to save user meta on custom admin page

update_user_meta($current_user->ID, 'description', $_POST['user_description']);

in the form processing function will save the bio.

I would keep the form action on the same admin page. As for data validation, I don’t see what there is to validate in the first place. It’s just a string – not an email, address or the like that needs to be checked against a certain format.