One the police are trying to arrest someone, whether through a warrant, exigent circumstances, or even by mistake, and the arrested party starts to flee, the police can enter any private property following the fugitive. This was a clear cut hot pursuit case.An actual arrest warrant allows them to enter the house upon opening the door.
Had he not identified himself, it would have made an arrest difficult, and they probably could not have entered the property. He identified himself, and then heard he was under arrest, before fleeing into the house.