How could the righteous Christ become sin for us?

Luther:

Christ was, in His suffering, completely righteous and completely sinner, completely saved and completely damned at the same time – and if we do not want to say it like this, then I do not see how He is forsaken by God (Psalms 22:1). I want to make this clear in the following way: Christ was indeed saved and He stayed that – ‘because He had done no violence, neither was any deceit in His mouth’ (cf. Isaiah 53:9). Because He wanted to be born of the virgin – conceived by the Holy Spirit -, so that He would be completely without sin – because how else could He have redeemed us from sin?

But when He suffered, He bore all our sins as if they were His Own sins. He has suffered what we had to suffer exactly identically for our sins – and what the damned now suffer. Paul also says this in Romans 15:3 – when he cites Psalms 69:10: ‘The reproaches of them that reproached Thee, fell on me’. And Isaiah says: ‘Surely He hath borne our griefs, and carried our sorrows: yet we did esteem Him stricken, smitten of God, and afflicted. Because He had done no violence, neither was any deceit in His mouth’ (cf. Isaiah 53:3 and 9).

This is the great secret of the Godly grace that is given to sinners through the suffering of Christ: that through a wonderful exchange our sins are no longer ours but Christ’s sins, and that the righteousness of Christ is no longer Christ’s righteousness, but our righteousness. For He hath made Him to be sin for us, who knew no sin; that we might be made the righteousness of God in Him (cf. 2 Corinthians 5:21). This secret cannot be spoken, preached, presented and made clear enough to anybody – because of the height and the depth of the riches and glory that are purposed there.