Here, Jesus is recognizing His body as a gift given so that the Father's will may be done. Animal sacrifices could not accomplish God's will, but the sacrifice of the sinless God-man, Jesus of Nazareth, could. It has the power to cleanse from sin so that a New Covenant, a whole new religious order, may be established based on a personal relationship—unparalleled in its intimacy—with our Creator.
A major weakness of animal sacrifices is their failure to produce a desire in the offerer to obey God. No animal life is equal in value to a human life. Though we may grieve at the loss of a pet, an animal's sacrificial death cannot have a real impact because it will not motivate us to do anything. But when a human dies for us, we feel it! We feel we owe something in return; indebtedness arises from our gratitude for what the sacrifice accomplished.
In our case, the most valuable Life ever lived was given. Gratitude, worship, and obedience are the only appropriate responses to such a sacrificial gift as the body of Jesus Christ. There is no other acceptable sacrifice for sin that will allow us to continue living.
The theme of Passover is the awesome cost of salvation, which is manifested in the sinless sacrifice of Jesus Christ. His was not a mechanical sinlessness, but He was sinless, innocent, even while encumbered with the frailties of human nature just as we are. His was sinlessness with sympathy, empathy, compassion, kindness, and concern for the helpless slaves of sin. Understanding this, we should feel revulsion that our sins caused such an injustice as His death to occur. At the same time, we should also express appreciation, indebtedness, and thanksgiving by departing from sin.