“But now hath He obtained — a more excellent ministry — by how much also He is the mediator of a better covenant — which was established upon better promises.” He.8:6
This is a description of our Lord Jesus Christ and His divinely declared ministry as a holy, heavenly and eternal High Priest/Substitute for His people. When it says that He “obtained a more excellent ministry” — it is telling us that the governor of all of providence has finally brought to pass the time for the manifestation of His eternal Son — as our eternal Savior Substitute and Mediator.
The Levite priests had an outstanding ministry — that they were divinely appointed to religiously perform according to the law of Moses. But, they acted only as a type of savior/substitute — for the Old Covenant children of God. But, the Lord Jesus Christ — the sovereign Savior/Son of God — manifest in human flesh — was divinely sworn to be the eternal, immutable and heavenly High Priest — whom God had promised would come into this fallen world and fulfill all that needed fulfilling — for all those whom He had covenanted with God the Father — to come into this world and save! Hence, it says: “He is the Mediator — of a better covenant”! Even a “New” covenant — that is — as it says — “established (or, legislated) — upon better promises”.
The word for “promises” — literally means: announcements, or assurances. And, they are said to be “better” — because they are announced and assured as being already fulfilled! 2Co.1:20 tells us: “For all the promises of God — in Him — are yea — and in Him — amen — unto the glory of God through us!” Christ has done it all — praise His Name! Indeed — Christ is — our all — and in all! (Col.3:13)
He.9:15 says: “And for this cause (Or, for this glorious reason) — He — is the Mediator of the New Testament — that by means of death — for the redemption of the transgressions…they (lit.) who have been called — might receive — the promise — of the eternal inheritance.” And, they receive it because they have been placed in Him! And 1Jn.5:11-12 plainly states: “And this is the record — that God has given unto us eternal life. And — this life — is in His Son! He that hath the Son — hath the life — and He that hath not the Son of God — hath not the life.” Hence, in the eternal Son of God — that is — in His person and work as the eternal Redeemer — the promised gift of eternal life — has been given — has been fulfilled — unto us by God.
Therefore, He.12:24-25 goes on to strongly substantiate this truth to us — by further saying: “And, to Jesus — the Mediator of the New Covenant — and to the blood of sprinkling — that speaketh better things than that of Abel (That’s because the blood of the Savior — speaketh better things than the blood of the saved one) — see that ye refuse not Him that speaketh (Or lit., see that ye deprecate not Christ). For if they escaped not who refused him that spake on earth — much more — shall not we escape — if we turn away from (Or, if we pervert) Him that speaketh from heaven!”