Charles Ellicott 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 word which is here rendered “offer” (asah), like the Greek poiein, means literally “do.” Its sacrificial meaning, however, here, is entirely dependent on the context.

By the dead. — That is, because of, or on account of the dead body with which he had been brought into contact.