The statement, sometimes rendered as "Rome has spoken; the cause is finished" or in Latin " Roma locuta; causa finita est," derives from a statement Augustine made early in the fifth century.
In a sermon to his flock, Augustine informed them that the pope had ratified the condemnations of the Pelagian heresy pronounced at the councils of Milevi and Carthage. He said "The two councils sent their decrees to the Apostolic See and the decrees quickly came back. The cause is finished; would that the error were as quickly finished (Sermon 131:10)." This has developed over the centuries into the commonly known formula.
Augustine was commenting on the authority of the pope and the fact that councils of the Church are authoritative only if approved by the bishop of Rome.