To authorize is to give official approval or clearance for something. Authorizing can also delegate power to someone.
Authorities are people with power, and to authorize is to use or grant some of that power to another. A visa authorizes you to visit a specific country. A search warrant authorizes the police to search a house. A doctor's note authorizes someone to miss work or school. Also, authorizing can be a form of delegating powers or responsibilities. The president of a company could authorize a worker to make a particular decision. Authorizing officially allows something to happen.