In the context of the sentence "They authorized a search warrant of his house and officers went to the premises to search it," the word "authorize" means to give permission to do something. In this case, the police officers were given permission by a judge to search the man's house. The police wanted to search the man's house, but they needed to ask a judge for permission first. The judge said yes, so the police went to the man's house and looked around. This means the judge authorized the search.

The verb "authorize" and the noun "authority" are related. An authority is a person in charge, like an officer of the law. When an authority grants permission for something, they are authorizing. More generally, "to authorize" means to grant permission, whether or not the permission is granted by an authority.