The Holy Spirit is to the Church what the soul is to the body. He is the Church's soul, the Church's life. The Church is "the Temple of the Holy Spirit" (CCC 737). Our bodies, too, are temples of the Holy Spirit (see Cor 6:19), because we are cells in the Body of Christ.
The Holy Spirit infallibly guided the Church's earliest ecumenical councils to formulate the doctrine of the Trinity -- including the doctrine of the Holy Spirit. Why do Catholics believe in the Holy Spirit? On the authority of the Holy Catholic Church, which teaches it. And why do Catholics believe this Church has infallible authority? Because the Holy Spirit, and not any human spirit, is her soul.
"[T]he Church's mission is not an addition to that of Christ and the Holy Spirit, but is its sacrament" (CCC 738).
Catholic Christianity:A Complete Catechism of Catholic Beliefs based on the Catechism of the Catholic Church by Peter Kreeft