There are historical references and archaeologial proofs from times contemporary to the compostion of the Bible. For example, the inscriptions in the Roman Catacombs, references to the Neronian persecution in Roman Historians like Tacitus and Trajan, and contemporary Christian historical documents like the Letters of Clement of Rome to the Corinthians, the Didache called the Teaching of the Twelve Apostles, and the Epistle of Barnabas. When we come upon the first prolific writer of Christianity - St. Ignatius of Antioch, we find all of the marks of Catholicism already in his letters - the ministry of Bishops, Priests, and Deacons governed by Church laws, the terming of the Church as Catholic, the real presence, the veneration of Rome's teachings, and so on.
How does one know that it was handed down intact, say in 875 or 1312AD?
Because the practices of the people termed Catholic Christians at those times as shown in historical documents, manuscripts and archaeological history are identical to the practices of the Roman Catholics in the west and the Eastern Orthodox in the East which are peculiar to them and not held by Protestants - the threefold ministry (Bishops, Priests, Deacons) and the subministers (Subdeacons, Acolytes, Exorcists, Lectors, Doorkeepers), the holding of General councils of Bishops presided over by the Legates of Rome, veneration of icons and images and relics, belief in transubstantiation and purgatory, etc.