I believe the answer is found in apostolic succession and Peter’s authority which can be found in the Bible. (Matthew 16:13-19 and also in John 21:15-17.)
Jesus changes Simon’s name to Rock appointing him as steward who would carry the keys of the kingdom. Jesus reference to the keys can be found in (Isaiah 22) and they have always represented the delegation of authority. Having the keys means he can declare certain things to be lawful or unlawful, that is to bind or loose. An example can be found in The Council of Jerusalem in (Acts 15) and all councils since. Binding as loosing were so common in Rabbinic law, and these powers Jesus transferred to his apostles and first to Peter. This position was dynastic in that it was continuing and was filled whenever vacated, requiring a succesor. The Catholic church which has the unbroken succesion of Peter and the apostles, therefor has this authority.
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