Why do we refer to the Eucharist as an "unbloody sacrifice"?
First, the Eucharist or sacrifice of the Mass is not a re-sacrificing of Jesus. Jesus offered himself once for all (Heb. 7:27, 9:28), so the Mass is called unbloody because Jesus does not suffer and shed his blood anew.
However, while Jesus suffers and dies only once, his sacrifice as a whole culminated in everlasting glory in the heavenly sanctuary and is made sacramentally present at every Mass.
How can this be? The Catholic Church describes the Eucharist as “the source and summit of the Christian life” (CCC 1324). It’s the source because the Eucharist enables us to make present and offer anew Jesus Christ’s one redemptive sacrifice of Calvary, which began with his Passion (CCC 1362-68; 1341). It’s the summit because the Eucharist is truly a foretaste of heaven, in which we partake of Jesus’ body and blood as heaven and earth become most profoundly one.
The Eucharist is a communion sacrifice, which means we partake of Jesus in a way similar to how the ancient Israelites ate the flesh of the Old Covenant Passover lambs. But the New Covenant Passover is much more profound, for there is only one Lamb—the Lamb of God who takes away the sins of the world (John 1:29, 35-36)—and eating Jesus’ body and drinking his blood provides eternal life, not simply emancipation from a mere human and earthly oppressor, i.e., Egypt (John 6:52-59; see 1 Cor. 13:23-34).
Forty days after his Resurrection, Jesus ascends in triumph to heaven, taking his seat at the right hand of the Father (Mark 16:19, Luke 24:50-52, Acts 1:6-11). In doing so, Jesus culminates his one sacrifice of Calvary in everlasting glory, fulfilling also the Old Covenant Day of Atonement/Yom Kippur sacrifices (see Lev. 16). That’s because Jesus takes not the blood of goats and calves but his own, and he enters into the heavenly sanctuary, not one made by human hands (Heb. 9:11-14). So Jesus is the high priest of heaven (Heb. 8:1-3; CCC 662-64, 1137-39), and he always lives to make intercession for us (Heb. 7:23-25, 8:1-3, 9:23-24), which means the eucharistic sacrifice has continuing atoning power for the sins we commit daily (CCC 1366).
Jesus’ one sacrifice is made sacramentally present and offered anew at every Mass according to the order of Melchizedek, i.e., under the forms of bread and wine (Gen. 14:18-20, Heb. 5:7-10, Matt. 26:26-29, Luke 22:19-20; see CCC 1333, 1355, 1544). Consequently, the words of the Lord’s Prayer—“Thy kingdom come, thy will be done, on earth as it is in heaven”—are nowhere more profoundly fulfilled than in the Mass, because heaven and earth are united in a most perfect way that further fosters the redemption of mankind.
For more information on the intimate connection between the heavenly liturgy/work of Christ and the sacrifice of the Mass, see Pope St. Paul VI, Credo of the People of God, 24-26.