The short answer is that we believe that Christ is really present in the sacrament of Communion. To understand this, we must remember what a "sacrament" is. The prayer book defines it as "an outward and visible sign of an inward and spiritual grace" (BCP 1979 pp. 357-361). In short, it is a physical thing, like bread, wine, water, oil, that God uses to share the presence and power of Jesus Christ, through the working of His Spirit.
Yet, sacraments are not magic. It's not like we believe that priests have the power to "call down God from on high" to do what we want. No human, not even a bishop, has the power to make God do anything or to define how God will act. Yet, God does have the power to define how He Himself will act. And He has promised, in Scripture, to work in certain ways when we do certain things (2Ch 7:14; Jer 18:5-10; Num 21:8-9; Isa 55:11; Mat 9:21, 18:18-20).
One of these promises is when Jesus said of the Last Supper that "This is my body... this is my blood..." (Mat 26:26-28; Mark 14:22-24; Luke 22:17-20; 1Co 11:23-34). Another promise related to Communion is "whoever eats my body and drinks my blood will have life in him" (John 6:54-56). We believe that when we meet together as the Church and share in the Lord's Supper, that Christ is present in a real and unique way, so that it is an actual "partaking" or "communion" with the Risen Christ (BCP 1979, pp. 859-860; 1Co 10:16-22; Luke 24:25-31; Mat 18:18-20). This is because the Risen Christ fills the whole universe and is available to be present in a special way in the bread and wine of the Lord's Supper (Col 1:15-17; Eph 1:22-23, 4:10).
In fact, along with St. Paul, we believe that to have the Lord's supper and not discern, or understand, that Christ is really present is to become guilty of "sinning against the body and blood of the Lord" (1Co 11:27-29). Therefore, when the priest or bishop consecrates the sacrament by saying the words of institution (This is my body... This is my blood), we believe that Christ becomes fully present in the sacrament for our strengthening. He becomes fully present, not because the minister "calls Him down", but because He has promised to be there.
Somewhere in the priest's Eucharistic prayer, there is also what is called the "Epiclesis". The Epiclesis comes from two Greek words for "upon" (epi) and "calling" (klesis), and thus epiclesis is where the priest calls upon the Holy Spirit to come upon the elements of bread and wine and make them into the presence of Christ. He usually says something like: "Sanctify [the bread and wine] by your Holy Spirit to be for your people the Body and Blood of your Son, the holy food and drink of new and unending life in him". In this action the priest is not making Christ "more" present in the sacrament, because Christ is already fully present whenever the words of institution are said. Instead, the Epiclesis is a recognition of how Christ is present: by the power of His Spirit. It is the Spirit that makes mere wine and bread alive with the presence of Christ, just as He made a virgin's womb alive with the presence of Christ (Luke 1:35).
There are about six models in Christian history of how Christ is present (or not present) in the sacrament. First, there is the view of the Roman Catholics that is usually called "transubstantiation", coming from the Latin words "trans" (meaning beyond) and "substantia" (meaning substance). Their view is that when the sacrament is consecrated, it goes "beyond substance", and ceases to be bread and wine in any way. It becomes fully the body and blood of Christ, and it only appears and tastes like bread and wine.
The second model is the Lutheran model, usually called "consubstantiation" (from the Latin word "con", meaning "with"). In this view, the presence of Christ is "with" the substance of bread and wine. The sacrament is fully the substance of bread and wine, and fully the substance of body and blood at the same time. The body and blood of Christ is somehow "with, in, and under" the elements of bread and wine. Thus Communion becomes fully bread and wine, and fully Christ, just as Christ Himself is fully human, and fully God, at the same time.
The third model is the "spiritual presence" model, usually held by Presbyterians and some others. In this model, it is not so much that Christ is present as that His Spirit is present in the sacrament. Instead of Christ coming down to be "in" the sacrament, the Spirit raises us "up" to where Christ is. In the sacrament, the Spirit connects us with the risen Christ and His flesh and blood in heaven. I have heard this called the "elevator" view of the sacrament, because in it we are raised with Christ, and connected with Him in heaven.
The fourth model is called "receptionism". This is the model that Christ is only present if we receive Communion in faith and believe He is there. If we don't have faith, then He is not really there. This would seem contrary to Scripture, which says "whoever eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty of sinning against the body and blood of the Lord. A person ought to examine themselves before they eat of the bread and drink of the cup. For anyone who eats and drinks without recognizing the body of the Lord eats and drinks judgment on himself. That is why many among you are weak and sick, and a number of you have died. But if we judged ourselves, we would not come under judgment" (1Co 11:27-31). Whether or not we want Christ to be present, He is there in the sacrament. If we come in good faith and have confessed our sin, the presence of Christ strengthens us. If we come without faith and with unclean consciences, the same presence of Christ will discipline us, and even harm us.
The fifth model is the "absence" model, where Christ is absent from the sacrament entirely. In this model, Christ and St. Paul are interpreted symbolically. When Christ said "this is my body", he really meant "this represents my body". Thus, in the absence model, Communion is not really a "communion with Christ", because it is just a symbol that He is not present in. No grace is given. Nothing happens for us except remembering what Christ did for us (which could be done by looking at a cross, without going to the trouble and mess of serving a meal). In the absence model, the Lord's Supper is just a memorial to Christ, kind of like a grave stone.
The sixth model is the Anglican model, called "real presence". Actually, "real presence" is the lack of a model. It is merely a statement that we believe that Christ is really present. We don't know exactly HOW He is present (other than by the power of His Spirit), because it is a mystery. But we do know that He is present as He promised. "Real Presence" does not work with receptionism or absence views, but it does work with the other models. You will find Anglicans who believe in transubstantiation, consubstantiation, spiritual presence, and who refuse to specify any model, but we all agree on the "real presence" of Christ.
Now the question arises: what does the real presence of Christ do in the sacrament? At least eight things:
• First, it connects us with Christ and is a "participation" in His life (1Co 10:16-17; 2Pe 1:4). It actually draws us up into His divine life and allows that life to flow through us, just like plugging an electric cord into a socket. Through communion, the suffering, crucified Christ, and the risen, victorious Christ, are made known to us, right here, right now (Luke 24:24-32). It opens a "portal" of tangible, touchable, intimate connection with the historic, physical Jesus Christ.
• Second, it assures us of forgiveness, because it is the "blood of the covenant, which is poured out for many for the forgiveness of sins" (Mat 26:28). By participating in communion, we are participating in the action of our forgiveness, and accessing the blood by which we are forgiven. By coming into contact with the real presence of Jesus' crucified and resurrected body, we eat and digest the life giving flesh of the Messiah. In this, it is a medicine for our souls and bodies, which we take to spiritually strengthen us with the very presence of Jesus Christ.
• Third, it strengthens us for service. Jesus says that His flesh is "real food", and His blood is "real drink" (John 6:54-56). Physical food and drink are used for one main purpose: to strengthen our bodies to work and live. This spiritual food strengthens our spirits to work and live for God. It is a meal of grace and power to recharge our spiritual batteries.
• Fourth, it connects us with the Body of Christ across the world. Just as the elements are called "the body of Christ", so also the Church is called "the body of Christ" (1Co 11:23-12:27). When two or more Christians are gathered for Communion, they united not only to Christ, but to each other as well, as the Spirit unites them with the celebration of Communion as the focal point of that union (Mat 18:18-20; Eph 4:4-6).
• Fifth, not only does this meal unite us to each other and to Christ now, but it unites us across time with the communion of saints, the great cloud of witnesses that surrounds us (Heb 12:1). Through communion, we are present with Christ and the disciples in the upper room at the last supper. Through communion, we are present at the great heavenly wedding feast that we will eat on the last day (Mat 26:29; Luke 14:15-24).
• Sixth, it remembers the act of Christ's sacrifice on the cross (1Co 11:24-25). It serves as a memorial that reminds us, and a ceremony that re-enacts and re-presents for us, every Sunday, what Christ did on the cross. God knows that we worship with our eyes, hands, smell, and taste, not just with our ears, and therefore Communion is a re-presentation and a representation of Christ's cross.
• Seventh, it proclaims the hope we have in the resurrection and second coming (1Co 11:26). It is a triumphal declaration of the victory of Jesus over all the powers and principalities that stand against God's Love. It is a foretaste of the glory, power, and community that we will have in Christ's presence forever.
• Finally, it is a non-bloody sacrifice in which God sacrifices Himself for us, and we sacrifice ourselves to Him. It is not a repeat, or "do over" of Christ's sacrifice, which is "once and for all" (Heb 9). Rather, it is a re-enactment and re-presentation of Christ's one time sacrifice, in which His eternal sacrifice is made physically present to us right here, right now. In return, we offer the only sacrifice we can give to God: ourselves. We return a sacrifice of our selves, our praise, and our thanks in gratitude for what Christ has done for us (Rom 12:1; Heb 13:15-16).