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Penal Substitutionary Atonement is a doctrine that has been popularized under that name by John Stott in his book, The Cross of Christ. But the roots of this idea run deep. All the way back to John Calvin and Anselm of Canterbury. It is the idea that Christ had to die to satisfy the wrath of God.
The Western Church has had a long history of holding a more juridical (that is, "legal") idea of the atoning work of Jesus. According to this view, we are debtors because of sin and our sins are recorded in "the book!" We therefore owe God a debt. We have sinned against his glory and grace and are due punishment because we have dishonored Him. God, in His mercy, sent Christ to take the punishment for this debt upon Himself, satisfying the "accounts". While there is an aspect of that I may not dispute. It has its problems, doesn't it? This view, if not balanced, makes God out to be rather sadistic in the estimation of some.
Meanwhile, the view of the atonement taken by the Eastern Churches is far less juridical and more therapeutic. According to that view, Christ died to overcome death for our sakes and heal our nature so we can be reunited to God. In this way, Christ has redeemed in us the divine nature and released us to rediscover and require it.
I would propose that both are true. In particular, I would say that Penal Substitutionary Atonement is necessary, but not in the way you are used to thinking of it. As we are used to thinking of it, it is God's ledger that must be reconciled. And while I do not deny that God has a "ledger" (what, after all, is hidden from His eyes?), God can blot out His ledger at will. "Blessed is he whose transgression is forgiven, whose sin is covered. Blessed is the man to whom the LORD does not impute iniquity."1 Rather, it is our ledger that must be satisfied.
The ledger that needs to be satisfied is ours. The demand for atonement is a human problem rooted in our nature, in our neurobiology, and brought about by sin and the trauma it induces upon the human person. We have a bio-spiritual mechanism that acts like a ledger that keeps track of everything–including our trespasses, great and small! The same neurological mechanisms that locate us in time and space and keep track of our actions mark our faults and trespasses. Our brain keeps a ledger of faults. Try as we might to dismiss or forget them, they remain permanently fixed in some region of the brain. And the chemical mark of those micro-traumas or major traumas, expressed in stress hormones, is imprinted on our bodies.
This record of our sins demands restitution. Payment. This is why we often feel we need to make amends to God - to make some sort of payment or vow to "make it right". Our internal ledger seeks reconciliation in ways it cannot provide. The debt is permanently recorded in neurobiological processes and mechanisms. Only an outside party, only God, can make payment.
This is a new and very real way of understanding the atonement, rooted in neurobiology, not just theology. What it means is God made atonement not first for His sake, to satisfy His wrath, but to satisfy our internal ledger. Our faults are marked inwardly, and God knew we were stuck in those processes and needed to be made whole. Christ came to reconcile our ledger. To forgive the record of sins kept by us, not just God. We call it guilt. But guilt has a real neurobiological underpinning.
The expiation procured for us on the cross and by His resurrection work because it is the only sacrifice we can believe is great enough to satisfy our sins and reconcile our ledgers. And when we accept this by faith, it speaks to the emotional brain, the part of the brain that is scandalized by sin. The rational mind may make excuses, but they are never good enough for the emotional brain. Only Christ's sacrifice is sufficient to speak to the soul in this way; to go beyond the rational mind, to the emotional mind, to the place where the ledger is kept.
Finally, we can say that the Father and the Son are united in healing the human condition and making payment for our sins. The account to be settled is first ours, which stands in the way condemning us, blocking our way to God. And the payment made was made to heal our fallen nature precisely at this spot. The payment made is not first to satisfy God's wrath, but to make our nature whole again. To free us from recrimination and release the soul from bondage to the law and our own internal law.
[1] The New King James Version (Nashville: Thomas Nelson, 1982), Ps 32:1–2.