The reason Catholics interpret the Bible correctly is because the Catholic Church made the Bible. The Bishops of the Catholic Church decided which books were inspired by God, and hence, belonged in Sacred Scripture. This descion, of course, would require the Bishops to know and understand the teachings of the Apostle (Sacred Tradition), and it would require them to be under the guidence of the Holy Spirit, whom Jesus promised to lead His Church into all truths. If the Bishops did not have Apostolic Tradition, they wouldn’t be able to tell authentic gospels from false gospels and authentic letters from false letters. If the Bishops did not have Divine Guidence, they wouldn’t be able to tell which books were inspired by God and which books were the product of human piety. For example, the letters of Ignatius were almost included in the Bible - but the Bishops had decided to not include them in the Scriptures, because they were judged to not be inspired by the Lord. Ignatius’ letters are great letters, but they weren’t inspired by God.
Now, obviously, in order for the Bishops to know Apostolic Tradition, and to be able to tell which books didn’t include false teachings, they had to be able to intepret the words of Jesus and His Apostles. Otherwise, they could be easily fooled by, say, the gnostics, who used Christian figures and terminology to spread heresy. So the Bishops had to know about the two intepretations of Scriptures: the literal and the spiritual. And this interpretation had to come from the Apostles, who would know what they were talking about when they spoke and who would know what Jesus was talking about when He spoke. (Jesus taught and enlightened them to His teachings between the time of His Resurrection and His Ascension; during His ministry, the Apostles weren’t too bright and didn’t understood Him too much). Since the Catholic Church made the Bible, and Jesus promised the Holy Spirit would guide His Church into all truths, it would follow than that the Church would preserve the Scriptures - the verses and their interpetation - throughout the ages, by the power of the Holy Spirit.
But this dosen’t mean Catholics interpret the Scripture as they see fit. Catholics are not allowed to use personal interpretation; they may only interpet the Scriptures as the Church interprets them (i.e., according to the teachings of the Church). The Bible was made by the Church, not by indvidual Catholics. The teachings of the Church are the very teachings of the Apostles - the same teachings which the Church used to decide what books go in the Bible.