Good question. Many of the mefarshim take a particular approach in answering this question, and the Lubavitcher Rebbe puts them together in a sicha.
Korach specifically waited to bring his argument after they had reached Eretz Yisrael (and turned back due to the sin of the spies). Why?
Until now, Benei Yisrael were in a heirarchy. There was Moshe and Aharon at the top, closest to Hashem, who were outstanding in their greatness, their intellect, their character traits. It was due to this greatness that they deserved to be at the top. It was indeed of necessity, as the children of Israel were like innocent children, in need of the guidance of these leaders.
However, the point of coming out of Egypt was not to remain in the desert, learning Torah forever. In fact, that's not correct; the point of learning Torah is in order to actually live in the land, and perform the mitzvot, translating the learning into action and deed. And this is the point:
When it comes to doing a mitzva, every Jew is on the same level. I.e. Every Jew's mitzva is equally holy, and great.
Korach therefore was arguing that it doesn't make sense anymore to have a single, outstanding leader - what does that achieve? The destination of the Jews is to go and do the mitzvot, and in that state we will all be equal.
This argument comes out in what Korach was saying. "Why do you need a single mezuza, when you have a house full of books?" i.e. why do you need one leader, when you have a nation of leaders! "Why do you need a single strand of blue on your tzitzit if the whole garment is blue" i.e. if we are all holy, why do we need you to rule over us as "extra" holy? It's ridiculous and that's why he laughed.
He was wrong, as the lesson is that the mitzva itself is indeed equally holy for everyone, but the kavana that goes behind the mitzva is also of importance. Therefore, an outstanding leader like Moshe does indeed accomplish more for Hashem's plan when he does a mitzva.
Some mefarshim even say that Korach is right - but it's not the time yet. He was too early.