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) — finally and decisively purchased and secured the fulfillment of God’s promises for us.
It means that God, according to the new covenant promises, brings about our inner transformation by the Spirit of Christ.
And it means that God works this transformation in us through faith — 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 may the God of peace who brought again from the dead our Lord Jesus, the great shepherd of the sheep, by the blood of the eternal covenant, equip you with everything good that you may do his will, working in us that which is pleasing in his sight, through Jesus Christ, to whom be 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 accord with 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 under the tree, so to speak, for you to pick up in your own strength. He picks it up and puts it in your heart and in your mind and gives you the seal of assurance that you are a child of God.

– John Piper