In today's first reading we hear a sentence that many inside and outside the church would just as soon ignore.
There is no salvation through anyone else, nor is there any other name under heaven given to the human race by which we are to be saved.
This has been famously restated as "outside the Church there is no salvation."
Some have misrepresented this as a Catholic belief that only Catholics can get into heaven. That is not what the Church teaches.
The challenge comes in the fact that our faith as Christians is not primarily a philosophy or a moral code; it is not primarily grounded in a book. Our faith is centered in a person, Jesus. He is the fullness of God's revelation, the Only-begotten Son of God, the Word made flesh.
For any Christian the single historical act that made salvation possible was his death on the cross. This is how we understand Peter's statement.
The Sonship of Jesus is a fact, either true or not. As Christians we believe it to be true. Only when we die will we know, but for now I believe. As C.S.,Lewis puts it "Either this man was, and is, the Son of God, or else a madman or something worse." Given the claims of Jesus there is no soft middle position.
The thornier question is how do we participate in salvation.
The one sure way is baptism.
But that is not, we believe, the only way. Perhaps the most far-reaching statement is that found in our Dogmatic Constitution Lumen Gentium #16, "Those also can attain to salvation who through no fault of their own do not know the Gospel of Christ or His Church, yet sincerely seek God and moved by grace strive by their deeds to do His will as it is known to them through the dictates of conscience."
The Church as St. Paul tells us is the Body of Christ. If you are in any way attached to Christ, you must logically have some connection to the Church even if tenuous. As an adopted child I have no idea who my birth mother was, but because I am alive I am sure I had one and we are linked. Jesus's death is the event that opened heaven to human humans and therefore any human who makes it to heaven is linked to Jesus and his body, the Church.
That is what we mean by the controversial statement. It is simple logic.