Church as the Sacrament of Communion

Why is the Church called the "Sacrament of Communion"?

Because the loaf of bread is one, we, though many, are one body, for we all partake of the one loaf.
1 Corinthians 10:17

The Church is the Sacrament of Communion because of her unique and intimate relationship both within humanity and with God himself. The Church is fundamentally a community of men and women united in Christ's fullness of grace as Head of his Mystical Body. (Cf. CCC 1140)

The Church's communion, in Greek, koinonia, includes both an invisible dimension (intimate communion among the Persons of the Blessed Trinity and all humanity) and the visible dimension (communion in the teaching of the Apostles, the Sacraments, and in the hierarchical order).  This communion, then, implies a spiritual solidarity among the members of the Church inasmuch as they are members of one Body, united in Christ. The union of the Church with Christ is described in Scripture as that of a Bride with her Bridegroom.
(Cf. CCC 771-773)

This communion is above all a gift from God, a new relationship between the members of the Church and God that has been established in Christ is communicated through the Sacraments and also extends a new relationship of humans among themselves. It is through this communion that God seeks to gather all of humanity to him in order to draw them toward eternal salvation—the ultimate unity of the human race rooted in heavenly glory.
(Cf. CCC 775-776)

-The Didache Bible

Why is the Church called the Mystical Body of Christ?

As in one body we have many parts, and all the parts do not have the same function, so we, though many, are one body in Christ and individually parts of one another.
Romans 12:4-5

As a body is one though it has many parts, and all the parts of the body, though many, are one body, so also Christ.... Now you are Christ's body, and individually parts of it.
1 Corinthians 12:12, 27

The Mystical Body of Christ is a scriptural image of the Church drawn from the teachings of Christ and Saint Paul that illustrates her unity in Christ, her relationship to him, and the interdependence of her members.

The image of the Church as the Body of Christ is found chiefly in the Epistles of Saint Paul as they describe the relationship between the Church and Christ.  Saint Paul's understanding derived from his conversion on the road to Damascus, when Christ asked him: "Saul, Saul, why are you persecuting me?... I am Jesus, whom you are persecuting." (Acts of the Apostles 9:4-5; emphasis added). Christ had already completed his earthly ministry; undergone his Passion, Death, and Resurrection; and ascended into Heaven, but he identified the persecuted members of the Church on earth with his own Body.  During his public ministry, Christ also expressed this intimate relationship between himself and his Church, where the two are spoken of as one Body. (Cf. CCC 787)

The image of the Church as the Body of Christ is significant because it indicates that the Church is not simply a community of members gathered around Christ, but that the Church is united in him, in his Body. (cf. CCC 789) 

The faithful are incorporated and fortified into the Body of Christ through the Sacraments of Initiation, beginning with Baptism.  Through the Sacrament of the Eucharist, the faithful are united together in Christ's Mystical Body; that is, brought into communion with one another through communion with Christ, their Head.
(cf. Colossians 1:17-18) (Cf. CCC 789)

As the Mystical Body of Christ, the Church extends Christ's work of salvation throughout time. The faithful play diverse roles in the Church just as various body parts have diverse functions (cf. LG 7).  Rather than harm the body, this serves its unity. (Cf. CCC 776, 846, 1111)

The Holy Spirit acts in the Mystical Body of Christ by giving us grace, unifying and animating the Body, comparable to how the soul functions in the human person (cf. LG 7). (Cf. CCC 797,1108)

-The Didache Bible