"The central Christian belief is that Christ's death has somehow put us right with God and given us a fresh start."
Gutenberg Bible (Library of Congress)
GNU Free Documentation LicenseGutenberg Bible (Library of Congress) - Credit: Raul654

As he makes clear later (page 59) Lewis is talking about the doctrine of the atonement of Christ--the act by which Christ reconciled sinful men to God, allowing humans to be "at one" with God.

There are various views regarding the atonement. Lewis avoids arguing about the specifics of it, but the most widely accepted Protestant view is that Christ's atonement was "vicarious."

"The word 'vicarious' means substitute.  Therefore, Christ was a substitute for others in that he took their place and suffered their punishment.  It was also a legal act whereby Christ fulfilled the law and lawfully paid the penalty of sin." (Christian Apologetics and Research Ministry)

Bible passages that teach the substitutionary or vicarious atonement of Christ include the following.

Isaiah 53:1-6 (ESV): Who has believed what he has heard from us? And to whom has the arm of the LORD been revealed? For he grew up before him like a young plant, and like a root out of dry ground; he had no form or majesty that we should look at him, and no beauty that we should desire him. He was despised and rejected by men; a man of sorrows, and acquainted with grief; and as one from whom men hide their faces he was despised, and we esteemed him not.

Surely he has borne our griefs and carried our sorrows; yet we esteemed him stricken, smitten by God, and afflicted. But he was wounded for our transgressions; he was crushed for our iniquities; upon him was the chastisement that brought us peace, and with his stripes we are healed. All we like sheep have gone astray; we have turned—every one—to his own way; and the LORD has laid on him the iniquity of us all.

2 Corinthians 5:21 (ESV): For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God.

1 Peter 2:22-24 (ESV): He committed no sin, neither was deceit found in his mouth. When he was reviled, he did not revile in return; when he suffered, he did not threaten, but continued entrusting himself to him who judges justly. He himself bore our sins in his body on the tree, that we might die to sin and live to righteousness. By his wounds you have been healed.