The Korah story teaches us that great responsibility comes before great privilege.
You can't have a mass rebellion of 15,000 people, among them hundreds of community leaders, and fail to admit that something needs fixing. You can ascribe dirty political self-interest to Korach, you can doubt the sincerity of his motivations, but you can't deny that he led a movement, and a movement doesn't form without authentic concerns and feelings behind it.
It seems that there were no small number of people who felt lorded over by the Kohanim and Leviim, who had been given a great centrality and importance in the organization of the camp. In all likelihood, these feelings were reinforced by Kohanim and Leviim who felt and acted superior to the average Jew.
Of course, the choosing of the Levites actually had precisely the opposite intention, as does much of the legislation regarding the Kohanim. Korach's popularity highlights that these messages were not necessarily fully internalized, and so chapter 18 comes along to reinforce them once again.
Unlike in Spiderman, where first comes great power, and with it, great responsibility, chapter 18 emphasizes that the order is reversed. First comes the responsibility which the Korach fiasco left the Jewish people requesting- the responsibility to guide the Jewish people to a safe encounter with the Divine. The Leviim are responsible for the people, and the Kohanim are also responsible for the Leviim. In both cases, the text emphasizes that those who have been charged with the responsibility will be held personally accountable for any misdeeds on the part of the people who they are meant to guard.
It is only on account of their personal accountability for the people's sins that the Leviim have the privilege of receiving their gifts. The same goes for the Kohanim's role vis-a-vis the Leviim.
Only with great responsibility comes privilege.