John Gill Commentary Numbers 6:11

"and the priest shall offer one for a sin-offering, and the other for a burnt-offering, and make atonement for him, for that he sinned by reason of the dead, and shall hallow his head that same day." — Numbers 6:11 (ASV)

And the priest shall offer the one for a sin offering, and
the other for a burnt offering
That is, one of the turtles or young pigeons for the one kind of sacrifice, and one for the other sort; both being necessary; the one to expiate sin, and the other as a gift to God by way of thankfulness for acceptance of the former:

and make an atonement for him, for that he sinned by the dead ;
by being where the dead body was, which, though not sinful, in a moral sense, was, in a ceremonial one, and therefore required a sacrifice to atone for it; and which atonement was made by the sin offering typical of Christ, who was made an offering for sin:

and shall hallow his head the same day ;
consecrate himself to God afresh, particularly the hair of his head, let that grow again and begin his Nazariteship anew; so Jarchi interprets it, to return and begin the account of his Nazariteship.