Why did Moshe fall on his face?
Rashi: He fell on his face out of helplessness - due to the Machlokes. He prayed for Yisrael after the Eigel (Sh'mos 32:1), the Mis'on'nim (11:2) and the Meraglim (14:13). This was the fourth sin. He was afraid that Hashem would not accept his Tefilos on behalf of Yisrael.
Ramban (on Pasuk 5, citing Ibn Ezra) and Rashbam: To pray for guidance - and that is when Hashem told him what to answer Korach (in Pasuk 5). 1
Why did only Moshe fall on his face and not Aharon?
Ramban: Because, due to his deep humility, Aharon conceded that Korach was right, in that he was more fitting to be Kohen Gadol than he (Aharon) was. 1
Refer to 16:4:1:2,3.
Ramban: Which explains why, although Aharon followed all Moshe's instructions, he did not utter a word for the duration of the rebellion.
QUESTIONS ON RASHI
Rashi writes that he fell on his face, for this was the fourth sin. He fell also at the Meraglim!
Sifsei Chachamim: At the Meraglim, he fell and prayed. Here, it does not say that he prayed, so Rashi needed to give a reason why he fell.
Rashi writes that he fell on his face, for this was the fourth sin. What is special about the fourth?
Moshav Zekenim citing Bechor Shor: It is due to "Al Sheloshah Pish'ei Yisrael v'Al Arba'ah Lo Ashivenu" (Amos 2:6).
Ner Uziel (p.180-183): Chet ha'Egel involved all three primary bad Midos