Numbers 6:5 - 'Until the time is completed for which he separates himself to the Lord, he shall be holy.'
What does it mean to be holy in the eyes of the LORD?
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Rashi (a preeminent biblical commentator) explains that this verse refers to the hair, not the Nazir.
["it" not "he"]: his hair shall be holy inasmuch as he must let grow freely the hair of his head
You are correct however that a Nazir is holy as it says in Numbers 6:8 (transl. Artscroll)
All the days of his abstinence he is holy to Hashem
In Judaism, holy often has a connotation of separate. A Nazir separates himself from cutting his hair, wine and dead people. By doing so, a Nazir aims to separate himself from the mundane by observing a meticulous lifestyle to get closer to God.
For more see this Chabad article.