This portion of the liturgy is sung immediately before the Sacrament of the Lord's Supper. It is an interesting placement for these words considering it is inspired by John the Baptist's words immediately before he baptizes Jesus (baptism being another sacrament).
These final words that we vocalize before receiving forgiveness are repetitive, yet profound. We name Jesus as the Lamb of God (just as John the Baptist does). We acknowledge that He takes away the sin of the world (just as John the Baptist does). We ask for His mercy. We ask for His peace. That mercy and that peace is grounded in the reality that Jesus is the one who takes our sins away. He takes away the sin of the world on the cross. He takes away the sin of all those gathered around the altar as they partake of His body and His blood for their forgiveness.
These final words spoken before the meal begins are like the music that is played to introduce an baseball player or a professional wrestler. These words and this music introduce Jesus as He comes down and dwells with us in bread and wine. The Agnus Dei is Jesus' entrance theme.
And much like Jesus, it is gentle, humble, and powerful. And it leaves us with what we ask for - His mercy peace.