Christ did this so that we can serve God. Thus, in order for us to serve God personally, we must be close to Him. Sin separates! What does sin do to relationships, either with humans or with God? It divides. When a person steals from another, do they become closer? If a spouse commits adultery, does that bring a married couple closer? No, it drives them apart. If a person covets something belonging to another person, does their relationship blossom? Sin separates.
Above all, it separates us from God. How can we be close to Him as long as we are sinning? Something had to be done, first of all, to bridge the gap: The sins had to be forgiven. Therefore, Jesus Christ, when He qualified by being blameless, voluntarily offered Himself to be the sacrifice that would overcome the division.
Before He did this, knowing He would die, He made out a will. He said, "When I die, those who take advantage of My death will inherit what I have inherited." The inheritance is to be in His Family! With it goes all the other promises: the promises of the Holy Spirit, eternal life, all the gifts, continual forgiveness, etc.
Whatever is needed, He will supply it. He will continue to stand between God and us, for a priest is one who bridges the gap between different parties to bring them together. He is saying, "When I am resurrected, I will always stand in the gap and be there when you need Me, and I will administer the Spirit of God."
Being brought close to God not only enables us to serve Him, it also enables the Father to serve us. Because we are in His presence, He can distribute to us the gifts that enable us to continue. Christ, then, is shown to be the Sacrifice for forgiveness of sin; the Mediator of peace between God and us; the Testator who died, passing on the benefits to us. These benefits work to remove the flaw, allowing us to keep the terms of the New Covenant.
We can then have a sustained and wonderful relationship with God. We can have His laws written on our hearts (Hebrews 8:10) and so be transformed into His image, qualified to share the inheritance of the promises with Him because we are like Him.