We draw a lot from the understanding of the succession the early Church fathers. I’ll quote a few. For example, Clement of Rome in the late first or early second century, when dealing with a dispute in Corinth in which the Corinthians tried to sack and replace their priests and bishops:
The apostles have preached the gospel to us from the Lord Jesus Christ; Jesus Christ [has done so] from God. Christ therefore was sent forth by God, and the apostles by Christ. Both these appointments, then, were made in an orderly way, according to the will of God. Having therefore received their orders, and being fully assured by the resurrection of our Lord Jesus Christ, and established in the word of God, with full assurance of the Holy Ghost, they went forth proclaiming that the kingdom of God was at hand. And thus preaching through countries and cities, they appointed the first fruits [of their labours], having first proved them by the Spirit, to be bishops and deacons of those who should afterwards believe. Nor was this any new thing, since indeed many ages before it was written concerning bishops and deacons. For thus says the Scripture in a certain place, I will appoint their bishops in righteousness, and their deacons in faith.
Chapter 44. The Ordinances of the Apostles, that There Might Be No Contention Respecting the Priestly Office.
Our apostles also knew, through our Lord Jesus Christ, that there would be strife on account of the office of the episcopate. For this reason, therefore, inasmuch as they had obtained a perfect fore-knowledge of this, they appointed those [ministers] already mentioned, and afterwards gave instructions, that when these should fall asleep, other approved men should succeed them in their ministry. We are of opinion, therefore, that those appointed by them, or afterwards by other eminent men, with the consent of the whole church, and who have blamelessly served the flock of Christ, in a humble, peaceable, and disinterested spirit, and have for a long time possessed the good opinion of all, cannot be justly dismissed from the ministry. For our sin will not be small, if we eject from the episcopate those who have blamelessly and holily fulfilled its duties.