God alone has authority to forgive sins (CCC 1441) However, Jesus gave the authority to priests to act in His behalf in the Sacrament of Reconciliation. CCC 1461 says "Since Christ entrusted to his apostles the ministry of reconciliation, bishops who are their successors, and priests, the bishops’ collaborators, continue to exercise this ministry. Indeed bishops and priests, by virtue of the sacrament of Holy Orders, have the power to forgive all sins “in the name of the Father, and of the Son, and of the Holy Spirit.” When the priests says “I absolve you…”, he is acting in the person of Christ (in persona Christi–CCC 1548).
Jesus gave this authority in a couple different passages. In Matthew 16 we see Him give authority to the Church (with Peter as the primary “steward”) when He says " I will give you the keys to the kingdom of heaven. Whatever you bind on earth shall be bound in heaven; and whatever you loose on earth shall be loosed in heaven.”". Then later in John, He breathed on the Apostles and said "Receive the Holy Spirit. 23 Whose sins you forgive are forgiven them, and whose sins you retain are retained.”
It was God’s will to allow the Church to have binding and loosing authority, thus we can be assured that He will honor the absolution given by priests.