Chapter 3 of M. L. Andreasen's book The Sanctuary Service provides a look at the priesthood in the early Jewish tabernacle services. In that chapter, Andreasen provides three prominent features of the work of the priesthood.
Mediation
The first of these features is mediation. The priesthood interceded for the sinner. The sinner must slay the animal, but the sinner could not place the blood on the horns of the alter of burnt offering or pour it out at the base of the same alter. The sinner could not go into the holy place and do the work there. The sinner needed someone to do something for the sinner in the Most Holy Place once a year. The sinner needed someone to mediate.
The lesson taught to Israel was that one would come who would intercede for humanity. That one would stand between humanity and the broken law and ultimately bring peace.
Reconciliation
The next feature is reconciliation. The sinner has had some distance placed between the sinner and God. However reconciliation was offered to Israel through the work of the priesthood. An atonement for forgiveness was offered to Israel every time they offered a sacrifice. (Leviticus 4:13, 20). Something about that work of the priesthood brought the gift of reconciliation to humanity.
Sanctification
The work of the priesthood also brought sanctification to humanity. We are able to be cleansed as a result of the work of the priesthood. Humanity may have sinned and been separated, but God is faithful and just to forgive and cleanse from all unrighteousness. (1 John 1:9).
Especially shown in the work of the day of atonement, the priesthood (High Priest) brought cleansing to the sinner as well as the tabernacle itself.
The priesthood brought to Israel mediation, reconciliation, and sanctification. Today our great High Priest in heaven seeks to provide these benefits to us in his great work for the salvation of humanity.