"Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: but this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people" (Jeremiah 31:31-33, King James Version).
God is just and holy, yet separated from sinners like us. This is our problem at Christmas, and every other season. The most critical question before us is: how shall we get right with a just and holy God? Had the festivities not become attached to Christmas, year after year, Christmas would have come and gone with even some Christians not noticing it. Nothing seems to have changed from the night God took birth on this earth in the person of Jesus. No one had time for Mary and Joseph. No one has time today to practise the teachings of Jesus.
A covenant is an agreement between two parties. In the past, God's people failed to keep their side of the covenant. Therefore, God, being merciful, decided to give us a New Covenant, Jesus Christ, the Messiah. God promised that someday He would do something new. He would replace shadows with the reality of the Messiah, and He would powerfully move into our lives and write His will upon our hearts, so that we are not constrained from outside, but are willing from inside to love Him, to trust Him, and to follow Him.
That is the greatest salvation imaginable — that God should offer us the greatest reality in the universe to rejoice in, and indwell us to see to it that we could actually rejoice in it, by granting us release from the clutches of sin and disobedience. That would be a Christmas gift worth singing about. But there was a huge obstacle: our sin.
How shall a holy and just God treat us sinners with such kindness as to give us the greatest reality in the universe, His Son, that we might rejoice with the greatest joy possible? The answer is that Jesus, God the Son, carried upon His shoulder the cumulative sin of humanity from the very first sin, culminating in the crucifixion of that sin upon the Cross, putting it to death, that sin which had bound us and distanced us from God. In this way, He deals with us mercifully while remaining just and holy at the same time. In Hebrews 9:28 we read: "So Christ was once offered to bear the sins of many; and unto them that look for him shall he appear the second time without sin unto salvation" (Hebrews 9:28, King James Version).
Christ has cancelled our guilt, and that means we can now be rid of our sins. When we lay hold of that, they no longer remain in God's mind as a basis for condemnation. In that sense, He "forgets" them. They are consumed in the death of Christ, and we are made free to live in adoration and worship of Him.