Why would God sacrifice His only Son for a fallen human race?
“For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.”**
Catechism of the Catholic Church:
604 By giving up his own Son for our sins, God manifests that his plan for us is one of benevolent love, prior to any merit on our part: “In this is love, not that we loved God but that he loved us and sent his Son to be the expiation for our sins.” God "shows his love for us in that while we were yet sinners Christ died for us.
620 Our salvation flows from God’s initiative of love for us, because “he loved us and sent his Son to be the expiation for our sins” (I Jn 4:10). “God was in Christ reconciling the world to himself” (2 Cor 5:19).
616 It is love “to the end” that confers on Christ’s sacrifice its value as redemption and reparation, as atonement and satisfaction. He knew and loved us all when he offered his life. Now “the love of Christ controls us, because we are convinced that one has died for all; therefore all have died.” No man, not even the holiest, was ever able to take on himself the sins of all men and offer himself as a sacrifice for all. The existence in Christ of the divine person of the Son, who at once surpasses and embraces all human persons, and constitutes himself as the Head of all mankind, makes possible his redemptive sacrifice for all.
478 Jesus knew and loved us each and all during his life, his agony and his Passion, and gave himself up for each one of us: “The Son of God. . . loved me and gave himself for me.” He has loved us all with a human heart. For this reason, the Sacred Heart of Jesus, pierced by our sins and for our salvation, “is quite rightly considered the chief sign and symbol of that. . . love with which the divine Redeemer continually loves the eternal Father and all human beings” without exception.