In this chapter, the LORD reaffirms his appointment of Aaron and the house of Levi with a miracle.
This is a relatively short chapter. In it, we can see the LORD wishes to finally put an end to the rebellions against Moses and Aaron, and he chooses to perform a sign before the people to reaffirm and prove Aaron's election as high priest. Honestly, it baffles me why the Israelites have not been convinced by everything they have already seen. Virtually everything that happened in Exodus centered around Moses in some fashion, and he continues to work miracles in their midst, such as bringing water from the rock. Why they continue to rebel is difficult for me to understand. And then in this chapter, simply by sprouting a staff, the people suddenly go from wanting to kill Moses to panicking that the LORD will slay all of them for their indiscretion (vv. 12-13).
Given the many rebellions, I think this reaction is totally understandable, I just don't get why it happens here rather than much earlier. I think there are a couple answers. The most significant we have to keep in mind is the public visibility (or lack thereof) for each miracle. That is, we are being given a full record of events by Numbers, but this information might not have been available to the ancient Israelites of Moses's time. Any miracle that went unwitnessed by the Israelites might as well have not happened for them. However, there have been many "global" miracles that should have been observed by nearly everyone. The plagues of Egypt, crossing the Red Sea, the manna from heaven, the thunder and lightning on Mount Sinai, the cloud by day and fire by night: all of these things are observable for the entire nation, and I believe this is intentional.
The LORD is trying to establish a relationship with the whole nation just like he did with Abraham. From this viewpoint, it is necessary for the LORD to use larger miracles to reveal himself to all the people because only larger miracles would be visible to everyone in a large geographic locale. Even so, a tremendous volume of the past couple books have been recountings of "the LORD told Moses", followed by "and then Moses told the people", which contributes to this problem that most people are disconnected from their god, and these many rebellions are the result of that disconnection. Since nearly everything is being told to the people indirectly through Moses, it's possible they question Moses's truthfulness when he reports his own promotion to leader and president of the entire nation. The miracles might be real miracles, but the only people saying "Moses is the leader" is Moses and his lieutenants. Also, don't forget the continual undercurrent of tribal politics: I'm sure many tribes are not happy to have a Levite, Moses, ruling over them and appointing other Levites into key religious positions.
The next point is that, while the people might have observed many miracles, it is possible they questioned whether those miracles affirmed Moses and Aaron as leaders of the people. I'm skeptical of this considering Moses's triumphal prominence over everything related to the LORD and the tabernacle, which strongly implies that the LORD favors Moses, but we should still keep in mind that this is really the first miracle that specifically and undeniably refers to Aaron and Levi.
The last point I will bring up is the potential weight of accumulated miracles. Maybe crossing through the Red Sea wasn't convincing enough, but now that they have a sprouted staff and some almonds, that is enough evidence for them to finally submit to Aaron's authority as priest.
And the Israelites' response is actually pretty funny. They must really be freaking out, since these two verses contain three different words for death: "gava", meaning to breathe out (your last) or expire, "abad", meaning to wander away, to be lost or perish, and "muth", meaning to die, to kill, to be destroyed. I would have expected something like this to show up during the plague for instance, when many thousands of people actually died. Only now do the people seem to expect that they will die, and this amuses me.
And that's the end of the story. After this we move back to a priestly legal section. While the story continues in chapter 20, this chapter concludes the current story arc, the rebellion arc. After this, the people generally stop challenging Moses's leadership, which suggests that they are truly afraid of the LORD killing them, and hence change their behavior for the better. Not to say that they stop sinning, but at least they stop questioning Moses and Aaron.