31 Therefore, as I said concerning the sons of Moses—for the sons of Moses and also the sons of Aaron shall offer an acceptable offering and sacrifice in the house of the Lord, which house shall be built unto the Lord in this generation, upon the consecrated spot as I have appointed—
32 And the sons of Moses and of Aaron shall be filled with the glory of the Lord, upon Mount Zion in the Lord’s house, whose sons are ye; and also many whom I have called and sent forth to build up my church.
33 For whoso is faithful unto the obtaining these two priesthoods of which I have spoken, and the magnifying their calling, are sanctified by the Spirit unto the renewing of their bodies.
34 They become the sons of Moses and of Aaron and the seed of Abraham, and the church and kingdom, and the elect of God.
I think the best way to understand this is in light of the language of the sealing to parents ordinance in the temple. The language there suggests that the effect of the sealing ordinance is to change the past, that it is “as if” the child were born within the covenant made to Abraham. This is in line with Abraham 2:10, in which God promises Abraham that: “as many as receive this Gospel shall be called after thy name, and shall be accounted thy seed, and shall rise up and bless thee, as their father.” Adoption, is, in essence, a rewriting of one’s birth, by changing who one calls mother and father, brother and sister. I don’t know that we are inclined to think of it as changing the past, but inasmuch as the titles mother, father, brother, sister all take their meaning from the moment of birth, to change who we call our family is in some sense to rewrite our past.
In light of this, I think D&C 33-34 are suggesting that as men come into the priesthood and magnify it, they become adopted sons of Aaron and Moses. Their bodies are “renewed;” it is “as if” their human familial relationships are rewritten. We use language similar to this when we say we are “reborn” at baptism and become the children of Christ.
In both cases, the change in family allows for greater opportunities and blessings. In the case of baptism, we can inherit all the Father has because we become the children of Christ, and therefore joint-heirs. In the case of priesthood, only those of Abraham’s family, or Aaron’s family, or Moses’s family, or of the chosen seed, have any right to the priesthood. By being adopted, we can thereby become eligible to perform the work associated with that family.