Why does the Church exist?

Answer 1

It exists so people have a place where they can worship God. It's a place where we learn about him.

Roman Catholic Answer

The Church exists because God established it! The Church exists because of God's great and inexhaustable love for us. The Church exists because It is the Mystical Body of Christ:

The Catholic faith emanated from God from the beginning of time, and was established on earth definitively from the side of Christ on the cross outside in Jerusalem when He side was pierced by a spear, around 33 A.D.. Then it was shown to the world at Pentecost with the coming of the Holy Spirit, fifty-three days later.

from The Catechism of the Catholic Church, second edition, English translation 1994

759 "The eternal Father, in accordance with the utterly gratuitous and mysterious design of his wisdom and goodness, created the whole universe and chose to raise up men to share his own divine life." (Lumen gentium 2) to which he calls all men in his Son. "The Father . . . determined to call together in a holy Church those who should believe in Christ." (Lumen gentium 2) This "family of God" is gradually formed and takes shape during the stages of human history, in keeping with the Father's plan. In fact, "already present in figure at the beginning of the world, this Church was prepared in marvelous fashion in the history of the people of Israel and the old Alliance. Established in this last age of the world and made manifest in the outpouring of the Spirit, it will be brought to glorious completion at the end of time. (Lumen gentium 2)

766 The Church is born primarily of Christ's total self-giving for our salvation, anticipated in the institution of the Eucharist and fulfilled on the cross. "The origin and growth of the church are symbolized by the blood and water which flowed from the open side of Christ as he slept the sleep of death upon the cross that there came forth the 'wondrous sacrament of the whole Church.'" Sacrosanctum concilium 5) As Eve was formed from the sleeping Adam's side, so the church was born from the pierced heart of Christ hanging dead on the cross. (Cf. St. Ambrose, In Luc. 2, 85-89: J.P. Migne, ed., Patrologia Latina {Paris: 1841-1855} 15, 1666-1668)

767 "When the work which the Father gave the Son to do on earth was accomplished, the Holy Spirit was sent on the day of Pentecost in order that he might continually sanctify the Church." (Lumen gentium 4; cf. Jn 17:4) The "the Church was openly displayed to the crowds and the spread of the gospel among the nations, through preaching, was begun." (Ad gentes 4) As the "convocation" of all men for salvation, the Church in her very nature is missionary, sent by Christ to all the nations to make disciples of them. (Cf. Mt 28:19-20; Ad gentes 2; 5-6)