There isn't really a "supposed to" or not when it comes to socks. You either prefer them or you don't.
Some wear them due to hygiene, some due to shoe sizing, and for some it's just comfortable.
Undoubtedly though, the majority of climbers do not wear socks with climbing shoes.
Most climbers prefer not to wear socks with climbing shoes, though, there is no rule against it.
Many climbers do in fact wear socks with climbing shoes for different reasons. Their shoes may be slightly too loose, and wearing socks could give them that extra layer that tightens things up.
Wearing socks could also help prevent blisters, or reduce pain if there are already blisters or other wounds on your feet as well.
So just because most climbers don't wear socks with climbing shoes, it doesn't mean there are no cases where it can be very helpful. Some people also just prefer sock in climbing shoes. It's up to you in the end.
