Christ has obtained a ministry that is as much more excellent than the old as the covenant he mediates is better, since it is enacted on better promises. —Hebrews 8: 6
Christ is the Mediator of a new covenant, according to Hebrews 8: 6. What does that mean? It means that his blood— the blood of the covenant (Luke 22: 20; Hebrews 13: 20)— purchased the fulfillment of God’s promises for us.
It means that God brings about our inner transformation by the Spirit of Christ. And it means that God works all his transformation in us through faith in all that God is for us in Christ.
The new covenant is purchased by the blood of Christ, effected by the Spirit of Christ, and appropriated by faith in Christ.
The best place to see Christ working as the Mediator of the new covenant is in Hebrews 13: 20– 21:
Now the God of peace, who brought up from the dead the great Shepherd of the sheep through the blood of the eternal covenant [this is the purchase of the new covenant], even Jesus our Lord, equip you in every good thing to do His will, working in us that which is pleasing in His sight, through Jesus Christ, to whom be the glory forever and ever. Amen.
The words “working in us that which is pleasing in his sight” describe what happens when God writes the law on our hearts in the new covenant. And the words “through Jesus Christ” describe Jesus as the Mediator of this glorious work of sovereign grace.
So the meaning of Christmas is not only that God replaces shadows with Reality, but also that he takes the reality and makes it real to his people. He writes it on our hearts. He does not lay his Christmas gift of salvation and transformation down for you to pick up in your own strength. He picks it up and puts in your heart and in your mind, and seals to you that you are a child of God.
John Piper. Good News of Great Joy: Daily Readings for Advent (Kindle Locations 446-447). Desiring God.