How did Jesus take our punishment by dying on the cross?

Jesus took our punishment of eternal death by sacrificing himself on the cross.

How?

This is the concept of atonement. A type of atonement is described in the OT, the Yom Kippur. On the Day of Atonement, the high priest slaughtered a bull and a goat. He sprinkled the blood against the mercy seat and put some blood on the horns of the altar.

There was a third animal. Lev 16:

21 Then he is to lay both hands on the head of the live goat and confess over it all the iniquities and rebellious acts of the Israelites in regard to all their sins. He is to put them on the goat’s head and send it away into the wilderness by the hand of a man appointed for the task. 22 The goat will carry on itself all their iniquities into a solitary place, and the man will release it into the wilderness.

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The live goat, the scapegoat, carried away all the sins of the Israelites.

Jesus fulfilled the Yom Kippur ritual by sacrificing himself on the cross to remove our sins. 1 Peter 3:

18 For Christ also suffered once for sins, the righteous for the unrighteous, to bring you to God. He was put to death [on the cross] in the body but made alive in the Spirit.

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Heb 9:

13 For if the blood of goats and bulls and the ashes of a heifer sprinkled on those who are ceremonially unclean sanctify them so that their bodies are clean, 14 how much more will the blood of Christ, who through the eternal Spirit offered Himself unblemished to God, purify our consciences from works of death, so that we may serve the living God!

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Lev 16 provided a horizontal reality of the atonement ritual. The true spiritual reality is in Jesus' sacrifice on the cross. He died for us to take away our sins so that we are not condemned to eternal death.

Romans 6:

23 For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord.

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