The "suffering servant" prophecy in Isaiah 52:13-53:12, with its picture of substitutionary atonement, is foundational for Christianity.
The Gospels indicate that this prophecy shaped Jesus' self-understanding. In Luke 22:37, Jesus says that Isa 53:12, which describes the servant as "numbered among the transgressors," would be fulfilled in him. Earlier that same evening he explained to his disciples that the wine of one of the cups of Passover represented "my blood of the covenant, which is poured out for many for the forgiveness of sins" (Mt 26:28). Here his language also reflects Isaiah 53:!2, which says that the servant "poured out his soul to death" and "bore the sin of many."
Jesus' disciples initially resisted the idea that he, the Messiah, would be put to death in Jerusalem (Mt 16:13-23), but they came to understand that Jesus indeed would be the suffering servant. Peter, who protested against this idea when he first heard it, later held up Jesus' example of sacrificial love as one for his disciples to emulate, using Isaiah 53 (1 Pe 2:18-25). In verse 22, Peter quotes Isaiah 53:9 ("there was no deceit in his mouth") in reference to Jesus. He makes several more allusions to Isaiah 53 in the ensuing verses.
Matthew sees Jesus' healing ministry as a fulfillment of Isaiah 53:4 (Mt 8:14-17).
Many people have come to faith in Jesus the Messiah through the powerful words of Isaiah 53. One example from the first century is the Ethiopian eunuch in Acts 8. Philip uses Isaiah 53 to explain the Gospel to the eunuch and lead him to baptism (Ac 8:26-40).
Paul recognizes in Isaiah 52:!5 a prophecy of the Gospel going to the nations (Romans 15:21.). John and Paul both saw in Isaiah 53:1 the fact that there would also be resistance to that revolutionary message (John 12:38; Rom 10:16). Paradoxically, Jesus' being "lifted up" on the cross would be an exaltation and a victory (Isa 52:14; John 12:32).
I recommend the book The Gospel According to Isaiah 53 for more on the amazing message of the Suffering Servant.