By whose stripes ye were healed (Isaiah 53)

So, as I said in my last post, Isaiah chapter 53 is considered by most Christians to be a prophecy of Jesus's death. (It is considered by many Jews to be a prophecy of the Holocaust, and I don't want to diminish that, because it's a very reasonable interpretation; the Holocaust was unimaginably horrible and the Jews of Europe paid a tremendous price for fascist antisemitism. I can definitely see that interpretation, and even the possibility--as I have said elsewhere--that this is a prophecy about two completely separate things. However, I'm going to focus on how this fits Jesus's death.)

Verse three says:

He is despised and rejected of men; a man of sorrows, and acquainted with grief: and we hid as it were our faces from him; he was despised, and we esteemed him not.

This is fulfilled in how Jesus was eventually rejected by his own people and sent to be crucified:

Mark 15:12 And Pilate answered and said again unto them, What will ye then that I shall do unto him whom ye call the King of the Jews? 13 And they cried out again, Crucify him. 14 Then Pilate said unto them, Why, what evil hath he done? And they cried out the more exceedingly, Crucify him.

Verses four through six talk about how Jesus died for the sin of the world, as summed up in 1 Peter 2:24 "Who his own self bare our sins in his own body on the tree, that we, being dead to sins, should live unto righteousness: by whose stripes ye were healed." Perhaps Peter is actually quoting Isaiah 53:5 here.

4 Surely he hath borne our griefs, and carried our sorrows: yet we did esteem him stricken, smitten of God, and afflicted. 5 But he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; and with his stripes we are healed. 6 All we like sheep have gone astray; we have turned every one to his own way; and the LORD hath laid on him the iniquity of us all.

Verses seven and eight:

7 He was oppressed, and he was afflicted, yet he opened not his mouth: he is brought as a lamb to the slaughter, and as a sheep before her shearers is dumb, so he openeth not his mouth. 8 He was taken from prison and from judgment: and who shall declare his generation? for he was cut off out of the land of the living: for the transgression of my people was he stricken.

This is quoted in Acts chapter eight:

Acts 8:30 And Philip ran thither to him, and heard him read the prophet Esaias, and said, Understandest thou what thou readest? 31 And he said, How can I, except some man should guide me? And he desired Philip that he would come up and sit with him. 32 The place of the scripture which he read was this, He was led as a sheep to the slaughter; and like a lamb dumb before his shearer, so opened he not his mouth: 33 In his humiliation his judgment was taken away: and who shall declare his generation? for his life is taken from the earth. 34 And the eunuch answered Philip, and said, I pray thee, of whom speaketh the prophet this? of himself, or of some other man? 35 Then Philip opened his mouth, and began at the same scripture, and preached unto him Jesus.

Verse nine:

And he made his grave with the wicked, and with the rich in his death; because he had done no violence, neither was any deceit in his mouth.

This is prophetic of Jesus's burial, as he died between two thieves but was laid in the tomb belonging to Joseph of Arimathea:

Matthew 27:57 When the even was come, there came a rich man of Arimathaea, named Joseph, who also himself was Jesus' disciple: 58 He went to Pilate, and begged the body of Jesus. Then Pilate commanded the body to be delivered. 59 And when Joseph had taken the body, he wrapped it in a clean linen cloth, 60 And laid it in his own new tomb, which he had hewn out in the rock: and he rolled a great stone to the door of the sepulchre, and departed.

The last three verses are reemphasizing that all that Jesus suffered was so he could be a sin offering:

10 Yet it pleased the LORD to bruise him; he hath put him to grief: when thou shalt make his soul an offering for sin, he shall see his seed, he shall prolong his days, and the pleasure of the LORD shall prosper in his hand. 11 He shall see of the travail of his soul, and shall be satisfied: by his knowledge shall my righteous servant justify many; for he shall bear their iniquities. 12 Therefore will I divide him a portion with the great, and he shall divide the spoil with the strong; because he hath poured out his soul unto death: and he was numbered with the transgressors; and he bare the sin of many, and made intercession for the transgressors.

As Paul says:

Ephesians 5:2 And walk in love, as Christ also hath loved us, and hath given himself for us an offering and a sacrifice to God for a sweetsmelling savour.

So all of that (and probably more) is why Christians believe this passage is about Jesus.