The Torah commands the nazir to bring a sin-offering on the day he completes the period he set for himself in his nazir vow. Is being holy a sin? Sefer HaChinuch explains: A person is a soul that is housed in a body, and Hashem expects a person to take care of his “house” so that his soul can function at its best. Pulling toward holiness afflicts the body; pulling toward sensual drives afflicts the soul. Although the nazir deemed it necessary to afflict his body through abstinence, perhaps he went too far, and for this he must bring a sin-offering.
Alternatively, Ramban suggests that once a nazir has conducted himself with purity, it is considered sinful for him to descend back to the impurity of normal life. One might ask: What did the nazir gain by acting like an angel and then reverting to being exactly as he was before? We may explain based on the well-known idea that in order to correct a negative character trait, a person has to conduct himself in the opposite extreme for some time, and then he should conduct himself in a balanced manner (Rambam, De’os 2:2). So too here, by practicing abstinence, the nazir rectifies his character, and he is then capable of returning to regular conduct without fear of sin.
In total, the nazir must bring three offerings, an olah offering that is entirely incinerated on the altar, a chatas sin-offering, and a shelamim offering that is partly eaten by the owner, partly eaten by the Kohanim, and partly offered on the altar. We may suggest that these represent the three stages of the nazir’s process: Firstly, during the period of his vow, the nazir dedicates himself to Hashem entirely like the olah. Then he descends back to normal life, for which he must bring a sin-offering. Now that he has attained a perfect balance between his body and soul, he celebrates with a shelamim that is shared by the altar, the Kohanim, and the owners, which allude to his now balanced character.
The nazir shaves his sacred hair and adds it to the fire beneath the pot of shelamim. He does not cast away his hair since that would suggest that he is rejecting his abstinence (Sifsei Kohen, cited in Otzar HaMidrashim). Instead, he includes the sacred hair in the offering of celebration over his newfound holiness. By acting like an angel, he has revealed the angel within.
