Thus the foundation of what we know as “apostolic succession” is established as one of the foundational acts of the Apostles in the Acts of the Apostles.
By the way, I am so glad that this book is named, “Acts” of the Apostles and not “the Preaching” of the Apostles or “the attempts of the Apostles to change the world” or even the “Amazing miracles” of the Apostles. The simple, bare “Acts” of the apostles shows they were men of action. They were pro active. They took action. They no longer sat around debating which one was going to be the best or the most wonderful master of the Kingdom of God.
They just got on with the action. They got on with preaching and teaching and governing the church, and this foundational action was to choose a successor to Judas. The man was to be one of those who saw Jesus, heard his teachings and was a witness to the resurrection.
Thus it has been for the last two thousand years. Jesus handed on his authority to the apostles: “As the Father has sent me, even so I am sending you. Go therefore and teach all nations. All authority is given unto you.”
The apostles then handed that authority on to their successors, and just as the original twelve were made up of a group of humanly speaking failures and flawed characters–as the original twelve were weak human beings whose only strength was their trust in God and his grace, so the successors of the apostles today are men who do the best they can and allow God to do the rest.
St Matthias went to martyrdom we’re not sure where and little else is known about him except that through him a historical foundation for the ministry of the church was established.
So if non-Catholics dispute apostolic succession, point to St Matthias. Peter as the chief apostle called the apostles to choose a successor showing that this important fact would remain as part of the church of which he was the Christ appointed Rock.