In the gospel of John (chapter 15 verse 20), our Lord gives the Apostles the principles and foundation of the authority that He was later to grant them. In doing so, He gives them both some words of encouragement, and some words of warning. He tells them that their authority will be based on His authority. His words are:
"Remember the word that I said to you, 'A servant is not greater than his master.'"
Thus, He already told them these things, and He wants to expand on it here. He is making it clear that they have a connection that is deeper than the normal connection between servant and master and He is about to explain what that means.
"If they persecuted me, they will persecute you".
Those who reject Jesus, will also reject the Apostles. Because His authority is extended to them, they should expect the same treatment He received. This is so because there is an automatic connection between Jesus and His Apostles. He grants them their authority, and so they are acting "in persona Christi".
"if they kept my word, they will keep yours also."
Those who genuinely believe what Christ has said, will also believe the Apostles. It is not a maybe, or just a possibility. And it should be noted, He does not say this merely as a fact (though it is a fact), He also says it as a promise; if someone is seeking to keep Jesus' words, then it is guaranteed that he will seek to keep the Apostles' words as well. To follow Christ, is to follow the Apostles.
It is this last statement that I want you to consider. "If they kept my word", Jesus says, "they will keep yours also". In other words, those who genuinely love and obey Christ, those who truly want to follow exactly what He said (regardless of whether they like it or not, regardless of whether it makes them comfortable or fits with what they have been told by others), these will follow the Apostles as well; and this clearly goes beyond those alive in the first century. To follow the Apostles is to follow all that they have done and said, all that they established and instituted. In other words, the Catholic Church is the Church which Jesus founded and was structured and set up by the Apostles at Jesus' direction. To obey the Catholic Church is to obey the Apostles.
Those who want to follow Jesus will end up finding and following the Catholic Church. Those who do not really want to follow Christ, will find something else. They may follow a sect or schismatic group, but they will not follow the Catholic Church. Jesus promises that His followers, will follow the Apostles. This is why people suddenly "discover" the Catholic Church. They want to follow Christ, and so He leads them to the One, Holy, Catholic, and Apostolic Church, because those who follow Christ will follow the Apostles. Therefore, the turning point for everyone of us is, and always must be: to desire to follow Christ (and not our own desires) and then we will be led to the Catholic truth. Those who turn from or try to corrupt the Church are turning from Christ.