While we do not know for sure who wrote the epistle to the Hebrews, we are quite sure we know why he wrote it. The Jewish believers to whom he wrote were in danger of returning to life and faith under the Old Covenant. In doing this they would reject Christ, and in essence depart form the living God (3:12). The author attempts to keep them from this horrible error by demonstrating the New Covenant’s superiority to the Old. Jesus, the mediator of the New Covenant is better than the Old Covenant functionaries including the angels (Heb.1&2), Moses (3), Joshua (4), Aaron (5) and the Levitical priests (7).
Jesus Christ and His priesthood are superior to the priesthood of Levi. The author demonstrated this in two ways in 7:20-25. First, unlike the priests who served after the order of Levi, Jesus was made a priest by God’s oath (7:20-22). Because of this, Jesus has become a surety of a better covenant. Second, unlike the Levitical priesthood, Jesus’ priesthood is unchangeable (7:23-25). Because of this, He is able to save completely and eternally those who come to God through Him. Those who came to God through the Levitical priesthood could not be perfectly and eternally forgiven of their sins. It was not possible that the animal sacrifices they offered could take away sins (10:4). Those who offered them by faith were ultimately forgiven when Christ offered the sacrifice to which they pointed. To return to the Old Covenant with its sacrifices was to reject Christ’s sacrifice, the only means of salvation, and therefore to be in danger of eternal punishment!

Of how much worse punishment, do you suppose, will he be thought worthy who has trampled the Son of God underfoot, counted the blood of the covenant by which he was sanctified a common thing, and insulted the Spirit of grace? (Hebrews 10:29)