The Catholic Encyclopedia puts it this way:
By voluntary submission to His Passion and Death on the Cross, Jesus Christ atoned for our disobedience and sin. He thus made reparation to the offended majesty of God for the outrages which the Creator so constantly suffers at the hands of His creatures. We are restored to grace through the merits of Christ's Death, and that grace enables us to add our prayers, labors, and trials to those of Our Lord "and fill up those things that are wanting of the sufferings of Christ" (Colossians 1:24). We can thus make some sort of reparation to the justice of God for our own offenses against Him, and by virtue of the Communion of the Saints, the oneness and solidarity of the mystical Body of Christ, we can also make satisfaction and reparation for the sins of others.
In layman's terms, that means that we believe that God is hurt and offended by sin and that we can console God and make up for those sins in some way by our prayers and sacrifices.
Just to clarify, We make acts of reparation as a gift to God, not as a service to particular sinners. If your mother was attacked by a mugger, you would rush to her side to console her. You’d go out and buy things to replace what was in her purse. You wouldn’t do that for the mugger’s benefit. Doing it wouldn’t lessen his culpability. Hopefully, you could also one day bring yourself to pray for him and wish for his conversion. But this would be a separate action than doing something to lessen the hurt of your mother right now.