“And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance.”
King James Version (KJV)
9:15 And for this end he is the Mediator of a new covenant, that they who are called - To the engagements and benefits thereof. Might receive the eternal inheritance promised to Abraham: not by means of legal sacrifices, but of his meritorious death. For the redemption of the transgressions that were under the first covenant - That is, for the redemption of transgressors from the guilt and punishment of those sins which were committed in the time of the old covenant. The article of his death properly divides the old covenant from the new.
Heb 9:15 He is the mediator of the new testament. "Covenant" here rather than "testament". The Greek word "diatheekee" means both "covenant" and "testament", but here "mediator" shows that "covenant" is referred to. That by means of death. His own death. For the redemption of the transgressions [that were] under the first testament. "The first covenant" (Revised Version). None under the old covenant could have complete redemption except by the death of Christ. Those who served God under it offered sacrifices which were types of Christ's sacrifices, but they could not be made efficacious without Christ. It is by his death that they who are called, of every dispensation, have the "promise of eternal inheritance".