Monergism

If I were go to your house and break your china, you may tell me that you forgive me, but in doing so you had to absorb the cost of the broken china. Someone has to pay for the broken china, either me or you, if you forgive me. The cost does not just go into oblivion. Likewise God does not wave his hands over sin when He forgave it, He must Himself absorb the full cost of the injustice done.
- Monergism

The view of Christ’s death presented here has frequently been called the theory of “penal substitution.” Christ’s death was “penal” in that he bore a penalty when he died. His death was also a “substitution” in that the was a substitute for us when he died. This has been the orthodox understanding of the atonement held by evangelical theologians, in contrast to other views that attempt to explain the atonement part from the idea of the wrath of God or payment of the penalty for sin. This view of the atonement is sometimes called the theory of vicarious atonement. A “vicar” is someone who stands in the place of another or who represents another. Christ’s death was therefore “vicarious” because he stood in our place and represented us. As our representative, he took the penalty that we deserve.
Wayne Grudem from Systematic Theology

The theory of penal substitution is the heart and soul of an evangelical view of the atonement. I am not claiming that it is the only truth about the atonement taught in the scriptures. Nor am I claiming that penal substitution is emphasized in every piece of literature, or that every author articulates clearly penal substitution. I am claiming that penal substitution functions as the anchor and foundation for all other dimensions of the atonement when the scriptures are considered as a canonical whole. I define penal substitution as follows: The Father, because of his love for human beings, sent his Son (who offered himself willingly and gladly) to satisfy his justice, so that Christ took the place of sinners. The punishment and penalty we deserved was laid on Jesus Christ instead of us, so that in the cross both God’s holiness and love are manifested. The riches of what God has accomplished in Christ for his people are not exhausted by penal substitution. The multifaceted character of the atonement must be recognized to do justice the canonical witness. God’s people are impoverished if Christ’s triumph over evil powers at the cross is slighted, or Christ’s exemplary love is shoved to the side, or the healing bestowed on believers by Christ’s cross and resurrection is downplayed. While not denying the wide-ranging character of Christ’s atonement, I am arguing that penal substitution is foundational and the heart of the atonement.
Thomas Schreiner

"Since Jesus Christ became a substitute for us all, and took upon Himself our sins, that he might bear Gods terrible wrath against sin and expiate our guilt, he necessarily felt the sin of the whole world, together with the entire wrath of God, and afterwards the agony of death on account of this sin."
-Martin Luther