The following summary represents what we find in the classics of early Christian thought as they recalled the 'faith once delivered,' and sought to articulate the meaning of the Incarnation in light of the revelation that Christ was both fully human and fully divine.
For primary readings on this, see for example:
Athanasius, On the Incarnation
Gregory of Nazianzus, Letters in Critique of Apollonarius
Cyril of Alexandria, On the Unity of Christ
When the apostles say Christ suffered and died for us, once for all (Rom 6:10; Heb 9:28; 1 Pet 3:18), for the forgiveness of sins (Acts 13:28; Col 2:13) and not ours only, everybody's (1 John 2:2), what does that actually mean?
1. The NT connects sin with it's inherent destructive consequences, its intrinsic judgment. Among the metaphors used for what sin holds over the sinner are 'wages' (Rom 6:23) or 'debt' (Matt 6:12). Having collectively turned from God -- our source of life -- to sin -- the source of death -- humanity has come under the domination of sin and it's bitter fruit.
2. The NT identifies the destructive consequences of sin, ultimately, as the curse of death (Rom 5:12; 6:23). Sin condemns us to 'perish' (John 3:16-18), a death sentence already at work in us, through which the satan holds us in bondage to fear all our lives (Heb. 2:15).
3. The gospel of Jesus Christ is that Jesus has come to rescue, redeem or ransom us from the curse of sin, which is death. The Incarnation was God's decisive redemptive act, through which he set us free from root to fruit: from the domination of sin, the corruption of fallen sinful nature, and the condemnation of death itself.
4. How does Christ accomplish this redemption?
a. The divine Word (God the Son) assumed the likeness of sinful human nature (Rom 8:3) in the person of Jesus Christ to heal human nature of the curse. As St. Gregory once wrote, 'Whatever is not assumed is not healed,' so Christ assumes the whole human condition in order to heal it all, including the curse of death itself.
b. Christ proclaims the Father's grace and freedom to forgive sin by freely forgiving sin throughout his life and ministry, and then does so once, for all and forever, when on the Cross he invokes the Father's forgiveness, even for the supreme human sin of deicide. The Father's answer comes through the voice of the Son, 'It is accomplished.' Our sin is forgiven and our lives washed clean by this act of mercy and grace.
c. Having freely forgiven us, we are reconciled to the Father, but the curse of sin must still be broken: death itself must be eradicated. So Christ does for us what we were unable to do for ourselves. He dies to enter death and so to overcome it. As all the church fathers testify (from Irenaeus to Athanasius, to the two Gregorys, Cyril and Maximus the Confessor) If Christ were merely God, he could not die. But if he were merely man, he could not defeat death. So Jesus Christ, fully God and fully man, enters death by death to annihilate death itself. This victory is made complete and manifest in the resurrection and ascension of Christ.
5. Thus, through Christ's incarnation, crucifixion and resurrection, he has brought about salvation (rescue from Satan, sin and death) for all people. His sacrifice was not the pagan appeasement of a wrathful deity, but rather, the sacrificial love of a God who became man to enter the human condition, including death and hades itself to rescue his beloved children.
6. Yes, Christ died for the forgiveness of sin, but as we see, this is an abbreviation that includes the truths that Christ came, lived, died and rose for the forgiveness of sin, cancellation of the curse, and defeat of death. Now we are invited to return to the open arms of the Father who opened the way back home through his Son. As we respond, we experience now what Christ already accomplished. By faith, we experience that forgiveness and freedom and salvation from sin and its awful consequences. We find that just as God in Christ participated in our human nature, we who are in Christ participate in his divine nature. As he took on our likeness to heal humanity, we are transformed more and more into his likeness and glory.
7. This is the apostolic testimony, received and faithfully preserved by the early church. This is not a theory of the atonement, but the gospel itself, the faith once delivered from the beginning. In this Gospel, Jesus is indeed a substitute, in that he does vicariously, as a man, what humanity could not do for itself. What is it that he does for us? God-in-Christ engaged and experienced the penalty (wages, debt or curse) of our sin -- namely death itself -- triumphing over it through his death and resurrection. In exchanging his life for our death, we rise with him in his life and find that death is no more.
For those committed to the language of 'penal substitution,' this telling of the gospel takes seriously the penalty of sin (death) and the substitution of Christ (as our vicarious representative), but it is distinguished from the much later version which identifies the penalty with God's wrath and punishment rather than sin's consequences and curse. In this telling, God the Word himself, via His incarnation as Jesus Christ, saves us from sin and death, swallowing them up in the magnificent victory of grace.