At first blush, and seen through the modern lenses of democracy and affirmative action, Korach's demands against Moshe Rabeinu do not seem so outrageous.
Keep in mind what's just happened. After having left Mt. Sinai for Eretz Yisrael, our ancestors stumble over and over again. From Kivrot haTa'avah to Kadesh Barnea, they commit one error after another, anger God on a repeated basis and get punished for it. The final one is the most drastic - they are told they will not be entering Eretz Yisrael but will wander back into the desert until they die.
Yet who gets exempted from that decree? Well, Moshe Rabeinu for one. And, curiously enough, his brother Aharon, the other Kohanim, Caleiv ben Yefuneh and Moshe's prize student, Yehoshua. From the point of view of the suddenly excluded, and with a cynical twist, that is enough to make one suspicious of the ruling power. Why is it that Moshe Rabeinu, his immediate family and friends all get to go to Eretz Yisrael while we have to die out here? And how come it's all those previously mentioned close family and friends that have all the biggest positions of power?
This was the point from where Korach began his agitations. Being a masterful demagogue, he tapped into the discontent that was now seething beneath the surface of the community and brought it to a boil. "Follow me," he promised the Bnei Yisarel, "and I will end this nepotism. This leadership is closed off to all but the well-connected and look what a lousy job they've done for us. I demand free elections! I demand transparency in the job selection process. I demand equal access to the Holy Sanctuary!"
As noted before, all tempting stuff for a people suddenly sentence to a very slow death in the desert.
A modern eye might look at Korach and his demands and conclude that he was onto something but that his approach was wrong. To do so would be a mistake because it would completely miss the point of Torah observance.
Rav Shimshon Rafael Hirsch, in his commentary on Chumash, notes that Korach's complaints revolved around the line "kedoshim tihiyu", "y'all shalt be holy" which starts Parshas Kedoshim. According to his interpretation, from that moment on all Bnei Yisrael were intrisically and equally holy. From Moshe Rabeinu down to the lowliest member of the community all were equal in the eyes of God. Thus what right did Aharon have to arrogate to himself the right to enter the Kodesh Kodeshim on Yom Kippur or designate the sacrificial service to his descendants? By what right was Moshe the permanent leader of the nation?
For Moshe (and God) however, holiness was not intrinsic. "Torah tzivah lanu Moshe", yes Moshe commanded us the Torah, but it remains "morashah k'hillas Ya'akov", an inheritance of the congregation of the Jewish people. Each individual Jew must, within that community, strive to obtain his part of the inheritance through the learning of Torah and the performance of mitzvos. Holiness is earned, not granted, in this approach which means there was a world of difference between the most holy and the least in the nation.
For Korach, it was about socialist equality of result, irregardless of effort. Simply showing up meant getting the prize. For Moshe, everyone was eligible to achieve holiness but they first had to make the effort. And which approach appeals more to the masses? Korach knew that the same lack of holiness that had spread itself through the Bnei Yisrael through their repeated stumblings made them ripe for his approach and ran with it.
In our day, this is exactly the approach used by the heterodox Jewish movements as well. One can be a "good Jew" simply by declaring that one's values are in consonance with one's personal view of what Judaism is. And since it's a sign of mental imbalance to declare a dissonance between the two, pretty much everyone who cares to be a observant Jew without going through the effort required to live a Torah lifestyle manages to achieve their own personal nirvana through the observance of a personal set of beliefs they call "Judaism".
Against this stands Toras Moshe. The Torah, as an expression of God's will, is an objective standard that we either measure up to or not. To be a good Jew is to make the effort to fulfill God's desires for us. Anything less is falling short of the mark and accepting such failure is catering to the lowest common denominator. This is why Korach was wrong and his followers punsihed so harshly. God demands personal excellence of us and gives each of us the ability to reach that level but we must step forward and make that effort ourselves. Anything less is not something we can accept.