I think its better if I explain the Catholic view of Matthew 16:18-19 first as it is the root of my faith.
18 And I say to thee: That thou art Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it.
19 And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose upon earth, it shall be loosed also in heaven.
First of all the naming of Peter as the Rock. Peter was born Simon. Christ gave him the name Peter when they first met. Peter is the English translation of Petros, which is the masculine of Petras, which means rock. So when Christ reiterates that Simon is Peter, as he did when they met, and then says upon this rock, he's referring to Peter. Its a pun. Now Peter would go on to become the first Bishop of Rome, and since the Pope is the Bishop of Rome that makes Peter the first Pope (even though the title of Pope had not yet been coined, the office is the same and the Pope today is also Bishop of Rome). Since Christ is God, and God is all knowing, Chirst knew that Peter would go on to become the first Pope when he named him, and because he knew that it means he was intentionally founding the Catholic Church in that moment. He also declared that the Gates of Hell shall not prevail against it, and since what Christ says is truth itself, that means the Gates of Hell shall not Triumph against the Catholic Church.
Line 19 about the keys to the kingdom of heaven and the powers to bind and loose clears up any doubt about what happened in line 18. It alludes to the end of Isiah Chapter 22 when Eliakim son of Helcias is called to be the new Steward of the Kingdom of David replacing the corrupt Shebna.
In other words, Christ is naming Peter as the Steward of his Kingdom on Earth, and granting him the necessary authority and power to minister the kingdom while Christ is in Heaven. Since Peter is a mere mortal, those powers become tied to his office and inherited by his successor as the Kingdom needs a Steward.
Because I believe Jesus Christ is the son of God, because I believe he is neither a liar nor a fool, I will hold true to the Church he founded because he said the Gates of Hell shall not Triumph against it.
Does that make sense?