Why did Jesus have to die? What did his death accomplish?

Clement of Rome fl. c. 96 ADFirst Letter to the Corinthians 49.6

Because of the love he had for us, Jesus Christ our Lord, by the will of God, gave his blood for us, his flesh for our flesh, and his life for our lives.

Christ's death as a substitute, "for us", in the first letter after the New Testament.

Letter to Diognetus c. 150 ADLetter to Diognetus 9.3–5

What else could cover our sins but his righteousness? In whom could we, lawless and ungodly, be made righteous, except in the Son of God alone? O sweet exchange! O work past finding out! O blessings beyond expectation! That the lawlessness of many should be hidden in one righteous man, and the righteousness of one should make many lawless righteous.

The "sweet exchange": our sin for his righteousness. Protestants often cite this passage. The verse just before says God "gave his own Son as a ransom for us".

Irenaeus of Lyons c. 130–c. 202 ADAgainst Heresies 3.18.7

What he appeared to be, he also was: God, summing up in himself the ancient formation of man, so that he might kill sin, destroy death, and give life to man. And therefore his works are true.

Irenaeus's "recapitulation": Christ goes over Adam's ground again, obeying where Adam disobeyed, and so undoes sin and death.

Origen of Alexandria c. 185–c. 254 ADCommentary on Matthew 16.8

To whom did he give his life as a ransom for many? Surely not to God. Was it not to the evil one? For he held us fast, until the ransom should be given to him for us, the life of Jesus, he being deceived and imagining he could master it, and not seeing that he could not bear the torment of holding it.

The earliest surviving statement of the ransom paid to the devil, on Matthew 20:28. Origen's point is the deception: the evil one took a life he could not hold, and death, having seemingly mastered Jesus, is mastered by him. The Matthew commentary witness in the store holds only books 10 and 11; book 16 is in Migne PG 13, from Khazarzar's second Matthew PDF (books 12-17), fetched for this dossier.

Athanasius of Alexandria c. 293/298–373 ADOn the Incarnation 9.2

Since the Word of God is above all, by offering his own temple and bodily instrument as a ransom for all, he fittingly paid what was owed by his death. And so, being united with all through a body like theirs, the incorruptible Son of God fittingly clothed all with incorruption by the promise of the resurrection. For the corruption of death has no more hold on human beings, because of the Word who dwells in them through the one body.

Death as a debt paid and as corruption undone, held together.

Gregory of Nazianzus c. 329/330–390 ADOration 45, on Easter 22

If the ransom belongs only to the one who holds us captive, I ask to whom it was paid, and why. If to the evil one, what an outrage, that the robber should receive not only a ransom from God but God himself as the ransom! ... If to the Father, first, how? We were not held by him. Second, why would the blood of the Only-begotten delight the Father, who would not even accept Isaac when his father offered him, but exchanged the sacrifice and gave a ram instead? Is it not clear that the Father accepts it, not because he asked or needed it, but because of the plan of salvation?

Gregory rejects a ransom paid to the devil and questions a ransom demanded by the Father. His answer: God accepted the sacrifice for our sake, to sanctify humanity.