Parshat Korach starts, of course, with the story of Korach. The story of Korach is basically a direct challenge to the authority of Moshe Rabbeinu. This is in contrast to the Sin of the Spies, which was a sin against Hashem directly, in that they did not believe in Hashem's power to act in this world.
Korach's challenge to Moshe's leadership was answered swiftly and forcefully with Divine Intervention. It started with Hashem saying that He would destroy the entire nation. Moshe and Aharon's prayers brought this down to the sinners themselves, Korach and his faction.
Nonetheless, the following day brought a complaint. The people blamed Moshe and Aharon for the death of Korach and his faction. That is to say that the people failed to recognize that the reason for yesterday's death and destruction was Korach's sin. This too brought a similar statement from Hashem to destroy the entire nation.
That is to say that the punishment was originally limited in scope, but then there were those in the nation that even after Korach's death did not recognize the reason for the outcome, and in effect, aligned themselves posthumously to Korach.
Moshe's argument in his first prayer (shall one man sin and You punish the entire nation?) was no longer relevant. The plague began. Moshe told Aharon to run quickly with the incense in order to stop the plague. Why?
Moshe and Aharon prevented destruction at first by their prayer to Hashem. It was not visible that they had done good for Am Yisrael. It went unnoticed. Therefore to fix this problem, Moshe and Aharon had to visibly do something for Am Yisrael.
- At first, the sin of the nation was in the heart (not yet verbalized), and therefore Moshe and Aharon's protection was not visible to the nation.
- When the nation followed Korach (so to speak) out loud, the protection of prayer was no longer sufficient.