What is the meaning of "Vayakatz Mo'av"?
Rashi: It means that Mo'av became sick of their lives.
What exactly, did Balak see that made Mo'av afraid?
Rashi (on Pasuk 2): He saw how the two kings upon whom they had placed their trust were unable to stand up to Yisrael, and he figured that that being the case, they would certainly not be able to.
Da'as Zekenim (2), Hadar Zekenim (2): Sichon defeated Melech Mo'av, and he could not stand up to Yisrael. All the more so I cannot!
Ramban: a. Because Yisrael was a far more numerous nation than Mo'av 1 , which was a relatively new nation, and b. because he heard about the ongoing miracles that Hashem had performed with them and with their forefathers.
Seforno: He saw how a. when Sichon refused to let Yisrael pdonkey through his land, they annihilated him, and b. how their victory did not come about in a natural way. 2
Seeing as B'nei Yisrael were not coming against Mo'av, Why were they distressed?
Riva: They were not afraid of Yisrael, for Hashem said "Al Tatzer Es Mo'av" (Devarim 2:9). They feared their surroundings
Rashi writes (Devarim 2:9) that Hashem forbade only fighting Mo'av; Yisrael appeared armed in front of them and took spoils, and therefore Mo'av feared them. I.e. Hashem forbade fighting them before the episode with Midyan. Really, Hashem forbade this after He said to take vengeance against Midyan, and Moshe thought that all the more so, to fight Mo'av (Bava Kama 38a)!
Riva citing R"M of Kutzi: Really, Hashem forbade fighting Mo'av before Ma'ase Midyan. Moshe thought that now that Mo'av began the clash through hiring Bil'am, we may fight them; Hashem refuted this.
Moshav Zekenim citing R. Tam: The Gemara means that Moshe would have made a Kal v'Chomer, had Hashem not already forbade fighting Mo'av. 1
Moshav Zekenim citing R. Elchanan: Bereishis Rabah (74) says that when Mo'avim told Yo'av that he may not fight them due to "Al Tatzer", he answered that the Isur was lifted after Balak hired Bil'am. This is unlike Bava Kama! Riva ha'Zaken