The Catholic Church claims teaching authority based on the argument that Jesus gave Peter, the mantle of the rock he will build his church. The Church argues it knows what Jesus meant, even though it is the first to admit it has often erred and has a history of ancient corruption and new corruption. The Church further argues that God cannot be in error, but our understanding of God is often in error.
As a Catholic, the arguments of the church are summarized in the Catechism which is available on the vatican.va website. The best apologetics I have ever read are those of Pope Emeritus Benedict and those of Peter Kreeft, a Catholic philosopher.
As for philosophical arguments, the teachings of the church are summarized by two arguments. a) We are all brothers and sisters, children of God, deserving of equal dignity and respect, and literally made in the image of God. b) We should reject evil even if it brings about good. This is a direct rejection of utilitarianism, the argument that the ends justify the means, and it is why the Church argues against eugenics, abortion, and capital punishment.
As for historical arguments, the Catholic narrative is like this. God intervened directly throughout mankind's history and has slowly revealed himself. First, to Abraham, then to Moses, and lastly through his divine Son. The world of ancient Europe during Jesus' time was dominated or some might argue enslaved by Rome. Through a divine miracle, the faith of impoverished uneducated Jewish fishermen became the official religion of the Roman and Byzantine empires.
The topic in question they are discussing is the status of Christ. So in this interpretation ‘This Rock’ refers to the statement Peter made about Christ in the previous passage. It’s simply Jesus asserting that this statement of Peter’s is true and the foundation of the Christian faith.
How are we to determine which interpretation is correct? It’s just a matter of opinion. There is no test we can apply to settle the matter. Therefore I think people fall back on other motivations to come to a conclusion - quite reasonably because I think they have no choice. They have to decide one way or the other somehow.
People generally don't choose a set of religious practices either. They are either born in that community or "convinced" to belong to it. People don't often have a choice between one or other "community and set of religious practices".
How do we tell correct inferences based on faith from faulty ones? Without that it's mainly just a lottery as to what faith your parents happened to have.
I concur with his contention that "pretending" to know what you don't know is not correct for faith. I just think people with faith are completely oblivious to the fact that their reasons are not what they think they are. After all they're only human and it's probably true for most of us.
I've become convinced that people generally don't choose a faith, they choose a community and set of religious practices. Usually that's the one most familiar or accessible to them, and this draw towards community is a powerful instinct. How many of us are capable of making rational decisions that would lead us to be ostracised by our community and the people we love?
Boghossian does sound like a bit of an arse though.