The USAM [US Attorney's Manual] requires such a search warrant to be approved by the US Attorney — the head of the office, a Presidential appointee — and requires "consultation" with the Criminal Division of the U.S. Department of Justice. This is not a couple of rogue AUSAs sneaking in a warrant...
Federal magistrate judges (appointed by local district judges, not by the President) review search warrant applications. A Magistrate Judge therefore reviewed this application and found probable cause — that is, probable cause to believe that the subject premises (Cohen's office) contains specified evidence of a specified federal crime.