The Church is a living continuation of Christ on earth because Christ spiritually and mystically lives and acts through the Church. It is a living organism, not just an organization.
Christ organized the Church to carry on His work in the world and to bring His ministry and truth to all men and all ages.
It is Christ Who accomplishes all that is done in the Church for man’s Salvation. The institution which is the Church survives because it comes from Christ, and because He promises to be with it to the end of time (cf. Matthew 18:20).
The Church is a living continuation on earth of its divine Founder. Jesus is indeed the Son of God, but He is truly a man too, the Son of Mary. Similarly, in the Church, the Mystical Body of Christ, is found the presence of Christ, its Head, and His Holy Spirit, the bearer of heavenly gifts. Yet, the Church is also very human. Its sublime mission is carried out by its baptized members. Among them are the ordained priests, who perform sacred ceremonies, called the sacraments, with ordinary realities of human life such as bread, water, wine, and oil. These ceremonies have been entrusted to the priests by Christ.
The Church acts for Christ. When a priest gives us a sacrament, it is Christ Who gives us the sacrament through the priest. When the Church speaks His word to us, it is Christ Who speaks to us and calls us to faith. The teaching and ruling authority of the Church is the shepherding of Christ.