If you give someone approval to do something, you are giving them permission. Think of the permission slip your parents sign to let you go on a field trip––they are approving your going on the trip.
Permission has the same Latin root word as permit. They both come from permittere, which means to allow to go or pass through. Permission can be given in writing, it can be given orally, or it can be assumed or unspoken––by passing the box of cookies around the room you're giving everyone tacit permission to help themselves.