The author of Hebrews is now describing how Jesus is not only a superior and ultimate mediator(priest) between us and Yahweh-God, but he serves as our go-between within a far better system of mediation.
In previous verses we saw that a temple, a special place that acts as a connecting point between us and God, is still a good thing. And it was implied in verse 2 that Jesus serves us in a better temple than the one that had been built on earth for worship of Yahweh.
The reason that Jesus is not serving as priest in the earthly temple is because that temple was set up for use by priests under the old system of law, who were only eligible as priests if they fit certain criteria, such as being born from the tribe of Levi. Under that system Jesus wouldn't even be eligible to serve as a priest, since he was born from the tribe of Judah. Jesus also offered a sacrifice completely different from those required by the old law. (Hebrews 7:27) In pointing this out, the author of Hebrews is making it clear that Jesus isn't "competing" with the Levitical priests for the honor of "greatest priest among priests". He is making them obsolete and replacing them.
The old system of priests operated within a building that, as ornate, and beautifully artistic as it was, only acted as a symbolic replica of the "true" tabernacle of Yahweh. The author of Hebrews calls the temple a "copy" and a "shadow" of the real thing. As the author points out, Yahweh had a very particular idea in mind regarding how the earthly tabernacle should be built. The reason for that revealed here is that it was meant to refer to the better tabernacle that would replace it.
Jesus serves us much more effectively than the old priests did, because his service, uniting us with God, is based on a better agreement ("covenant") between God and humanity. The old agreement for how relations would work between God and humanity was conditional. We had to be willing and able to hold up our end of the agreement or the whole system would fall apart. (Deuteronomy 28:15 and context, 28:36, 28:64) But this new agreement doesn't have any of the weakness of the old one, which is made more obvious when we compare the two.
The author says that God himself acknowledges shortcoming in the first covenant, and then quotes Jeremiah 31:31-34 to prove his point. Here, 600 years before Jesus was born, Yahweh tells the world in advance that he will be bringing about a new covenant that is better than the old one.
In the old one, Israel failed to hold up their end, so Yahweh was not obligated to carry out his end either. (v.9)
But under this new covenant, the law of God will be internalized on the hearts and minds of his people and the relationship between humanity and God will be restored.(v.10)
(It's worth noting that, although this covenant is with Israel, non-Jews who place their trust in Christ are "grafted in" to the family of God. This is hinted at in Genesis 12:3 and stated more clearly in Romans 11:11-27.)
Under this new covenant there will also no longer be a need for teachers, because everyone will know God deeply and will be constantly pursuing a deeper knowledge of and relationship with him.(v.11)
The wrath of God will be non-existent in this new covenant and no sins will be held against anyone.
In looking at this covenant, it's clear that not all of this has happened completely yet. But the process has begun:
Jesus "launched" the new covenant during his earthly ministry (Luke 22:20).
Paul also tells us that part of this covenant is in place now, in that God dwells within every believer, all of whom have become his "temple".(2 Corinthians 6:16)
And although teachers are still needed (Romans 12:7), the process of having God's law written on our hearts and minds has begun with the work of the Holy Spirit, God himself, living inside of all believers(John 16:13-15, Romans 8:1-7).
We are also seeing a change in how we relate to God. Obedience is no longer something we do to avoid God's wrath. Jesus has freed us from that hopeless endeavor. Now, obedience is something we do to express love, trust and gratitude toward God and enter into a more effective and purposeful way of living. (John 14:23-24)
And already, there is "now no condemnation for those who are in Christ Jesus"(Romans 8:1).
All of these things make it obvious that the old covenant is obsolete and ready to be replaced. (v.13)
Having said all that, it's still easy to fall into an "old covenant" way of thinking. When the world asks us to behave a certain way to fit in every day, it's easy to assume that if we've been distant from God for awhile we have to "shape up" and fall in line with his program before presuming that he will listen to us or care about our problems.
It's easy for me to think of my free time as "me time", not even bothering to invite Jesus to speak to me throughout the day, or giving him the freedom to interrupt my geeking out for something he wants me to suddenly think or pray about.
It's easy for me to box him in, as though his dwelling place is in my Bible, my study books or my church building instead of in my heart and mind throughout every moment.
But Jesus has freed us and is calling us out of that old covenant of moral obligation and compartmentalized living. He wants us to dwell with him and recognize that he dwells with us.
He wants to converse with us and celebrate the fun we're having with us. He wants us to recognize, in the moments of our geeking out, that he gifted someone with creativity in a way that he knew we in particular would enjoy. I can almost imagine him saying "That's for you! I KNEW you were gonna love that!"
He also wants us to be listening constantly, allowing him to suddenly interrupt our plans to engage in an opportunity he has orchestrated for us, to pray, to serve or to invest in someone else.
Some day the new covenant will be in full effect, but there's no reason not to begin experiencing it today.