In this chapter, Nadab and Abihu die after performing an unsanctioned ritual.
This chapter is basically a counterpoint to the last chapter. I described the previous chapter as a public vindication and endorsement of the priesthood, as we saw them offer sacrifices and the LORD came down and consumed (i.e. accepted) the offerings. Since the priesthood was very new, it was important that the people would see they had a divine mandate and not a human one.
However, that mandate came with a set of very strict rules governing the behavior of the priests and the form of their offering. We frequently saw the expression, "so that you do not die" in the text describing how the priests are to behave (for example, see Ex 28:35, Ex 28:43, Ex 30:20, Lev 8:35, and in this chapter, verses 7-9). This chapter teaches us that this was not an idle threat: if the priests do not obey the law, they can and possibly will die, as Nadab and Abihu die here.
The sin of Nadab and Abihu is not the idolatry of Ex 32; their sin is simply offering incense to the LORD in a way that is not prescribed by the laws given to them by Moses. Being burned to death seems like an overly harsh response, so I would like to remind my reader of three things. First, the priests were just publicly vindicated by the LORD. Second, this is the very beginning of the priesthood. And third, the rules given to Moses were meant to be followed exactly, just as the tabernacle was constructed precisely in accordance with the instructions given to Moses. Exodus goes to great length to tell us about this correspondence.
As a result, we should understand that whatever the priests do now, at the beginning of their ministry, will set the precedence for the priesthood to come after them. Since they were publicly vindicated, they are now in a position of notable authority, having the right to mediate sacrifices and atone for the people before the LORD. Since these two sons of Aaron were stepping out of line with their duties so early, the LORD had to take some sort of action to show that the priests do not dictate how they and the people relate to God, but rather the LORD dictates to them how they should relate to him. I think this issue largely comes down to power: is the LORD going to have power over the priests or are the priests going to have power over the LORD? The answer is obvious in hindsight. While the text doesn't say that they were intentionally challenging the LORD, I do feel that's one of the results of their actions. By not doing what the LORD says, the LORD is forced to either punish them or accept that they will not follow his laws going forward.
With all of that in mind, one could maybe insist that the LORD could just rebuke them rather than kill them. I think the proper response is that anything less than death would show a devaluation of the priesthood. The priests genuinely do have a lot of power, even when following the law, and the LORD wishes to make clear that this power comes with a lot of responsibility too. There are other areas where Moses rebuked the people for not following the law (for instance, with respect to the Sabbath), but after publicly approving the priests, he cannot now allow them to disobey him so quickly, acting as if they still had his approval even when doing things he had no commanded.
We see the pinnacle of this ferocious holiness when Moses commands Aaron and his remaining sons to not grieve their brothers "for the LORD's anointing oil is upon you." While before we saw the priests exalted and honored by their position, we can now see that it has a burden: even with the death of two of his sons, Aaron must go forward with the consecration process.
When the LORD commands Aaron to not drink wine, there is a principle of separation even between the priests and the people, just as there is a principle that separates the people of Israel (who are supposed to be holy) from the rest of the nations. Both of these principles are part of a grander rule, that to draw near to the LORD requires a separation from that which defiles. The people are a holy people because the LORD dwellls amongst them, and the priests are even more holy because they minister to the LORD in the tabernacle.
On a minor note, this is the first time I remember the LORD speaking directly to Aaron.
We are finally told in verse 14 that the priests' daughters (and by extension, wives, sisters, etc) may also eat of the wave offerings. They are further allowed to eat it in a "clean" place, which is more expansive than in the "holy" place that they can eat the grain offerings.
Lastly, Moses gets mad with Aaron's sons for not eating the sin offering (which we were told in chapter 6 that they could eat it, so long as its blood was not brought inside the sanctuary, in which case it had to be burned outside the camp). Moses is concerned they aren't following the law again, whether by accident or by intention. In this case, it's OK because they weren't commanded to eat the meat of the sin offering, it was their allowance. If they chose to burn it (as here), that's OK.
This concludes the (short) story section of Leviticus. The rest of the book is composed almost entirely of ceremonial laws. I hope you enjoyed storytime while you got the chance. :)