Scripture tells us of the history of God's self giving of Himself to us down through the ages which culminated in the ultimate giving of his Son. This Self-giving is at the heart of the relationship between God and ourselves. The Eucharist is the ultimate reminder of that self-giving which Jesus completed on the cross.
The concept of love as self giving was first set out with the coming of Jesus and the gift of the Eucharist extends that self giving of Jesus to all people for all time. And the Eucharist also gives us the greatest possible way of giving thanks for that gift.
The Eucharist ask us to give ourselves to others in the way as Jesus gave himself to us; selflessly and openly. It is not an easy command to obey, it is a very had thing to do. In the Eucharist Jesus gives us a model and reminder how we can give ourselves to others.