Most often this story is interpreted flatly, which is to say that we read as though Nadab and Abihu messed up, enraged God, and were struck down for doing something that was not permitted. Turning yet again to Rebecca Newberger Goldstein's magnificent work, 36 Arguments For the Existence of God: A Work of Fiction, which I am not so much reading as mining, she puts into the mouth of a Hasidic rabbi an interpretation of the strange fire offered by the sons of Aaron straight out of Jewish kabbalistic midrash, an explanation given by the Arizal (i.e., one Isaac Luria, a sixteenth century Galilean rabbi, mystic, and master of kabbalah- Arizal meaning Lion), which is a mystical interpretation, one that is at odds with more conventional interpretations.
The rebbe begins by citing the third verse of Leviticus chapter ten to the effect that Aaron was speechless. In typical English translations, it is stated even more clearly: "And Aaron held his peace." The rebbe goes on: Aaron's "silence was not only of words but of all reaction." Keep in mind this is in response to two of his sons, his heirs, being struck down! "Not a single tear crossed his cheek. Not a groan or wail escaped his lips," our interpreter says. The rebbe asks, "Was he speechless from horror? From grief?" "Maybe from self-protection, afraid to cross a line when, at that moment, the Judgment from On High had descended? Or was [Aaron's] the silence of an understanding that has answered its own question?" What does he mean here, to what is our Hasidic rabbi alluding? "What", the rebbe asks, "could have kept him from crying out after them?"
It is here that the rebbe pulls in Arizal "[i]n the last dr'ash that the Arizal gave before his death..." he compares the sons of Aaron "to the fawns of the gazelle." According to the Zohar, which, quoting Rabbi Joseph Telushkin, is "[t]he most famous work of kabbalah," with its origins in the thirteenth century, the gazelle "requires the serpent's bite in order to give birth." Based on this comparison, the rebbe, says, "Nadab and [Abihu] were...sacrifices, to hasten the coming of the Moshiach" (i.e., the Messiah). Hence, "[d]o not make the mistake of thinking that" the strange fire could be "idol worship." After all, the nephews of Moshe Rabenu (i.e., Moses our rabbi, the Lawgiver) would never give in to idolatry! So, on this kabbalistic interpretation, "[t]he strange fire was the redemptive fire that leaps out to purify the world, consuming the innocent only to return them back again in to the holy service, as it will always be, the gilgul turning round and round until the redemption of our days." He ends with, "may it be in our lifetime, Amen." For those of us in Utah, with our famous Gilgal Garden, it may be interesting to note that Gilgul neshamot is a kabbalistic term meaning something like "cycle of souls". As such, it is also something of heretical concept, as reincarnation is not found anywhere in Torah.
Of course, the more conventional reading of this astounding episode, is set forth by Robert Alter in his magisterial The Five Books of Moses. Alter tells us that Nadab and Abihu "would have filled the fire pans with glowing coals, not actual fire." He also points out that the adjective we transliterate as "alien," "strange" (i.e., zarah), or, in the case of the English Standard Version, to which I linked, "unauthorized," which is also an interpretation, one that certainly would not permit the kabbalistic interpretation given by the Arizal, likely means "unfit." Zarah, according to Alter, probably "indicates in cultic contexts a substance or person not consecrated for entrance or use in the sacred precinct, which is what prompts later translators to use "unauthorized." Alter further observes: "The consensus of of modern interpreters, with precedents in the classical Midrash," as opposed to mystical, that is, kabbalistic Midrash, "is that the fire is 'alien' because it has been taken from a profane source - e.g., coals taken from an ordinary oven."
Since the source of the coals is not given in the text, it remains an open question. It is the source of the fire, I think, on which the kabbalistic interpretation turns, as it could not be taken from a profane source and, as the rebbe points out, because it is unthinkable that Moses' nephews would engage in idolatry. The other axis on which the mystical interpretation turns is Aaron's peaceful response to his sons' being struck down. It is inconceivable to the rebbe that holy Aaron, the first high priest, would not be moved by sight of his sons being consumed by fire. But was it not Aaron's idea to fashion the ēggel hazâhâḇ, the golden calf (Exo. 32:1-4)?
Even today, for cultic (i.e., worship) purposes we do not light the sanctuary candles around the altar with a match, but with a beautiful brass taper, though we light the taper with a match.
In the words of John Cleese from Monty Python's Contractual Olibgation album: "And it came to pass that Saint Victor was taken from this place to another place... Here endeth the lesson."