What is the meaning of "Vayakatz Mo'av"?
Rashi: It means that Mo'av became sick of their lives.
What did Balak see that caused him to be afraid?
Rashi (in 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.
Ramban: He was afraid a. because Yisrael was a far more numerous nation than Mo'av, 1 since the latter 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 that, when Sichon refused to let Yisrael pass through his land, they annihilated him, and that their victory did not come about in a natural way. 2
Da'as Zekenim (in Pasuk 2) and Hadar Zekenim (in Pasuk 2): Sichon defeated Mo'av, and he could not stand up to Yisrael, how much more so he!
Ba?al ha?Turim: He saw that (during the battle with Sichon) the sun stood still for Moshe. 3
As the Pasuk specifically states.
Seforno (in Pasuk 3): 'And the strong men of Mo'av were afraid' (See Beshalach, 15:15), because they too, refused Yisrael passage through their land, as Yiftach told the king of Amon, in Shoftim, 11:17.
See Ba?al ha?Turim.
Seeing as Yisrael were not threatening to attack them, why was Mo'av afraid?
Rashi: Refer to Devarim, 2:9:1:1 and note.
Seforno: He saw how when Sichon refused to let Yisrael pass through his land, they annihilated him, and they too, refused Yisrael passage through their land (Shoftim, 11:17).
Riva: They were afraid not of what Yisrael 1 would do to them - since Hashem said "Al Tatzer es Mo'av" (Devarim 2:9), but of what they would do to their surroundings. 2
Perhaps they felt pain or awe due to the honor of miracles done for Yisrael, like they felt at the time of Keri'as Yam Suf (PF). Refer to Sh'mos 15:15:1:1 & 3.
What are the connotations of "Vayakatz Mo'av"?
Rashi, Rashbam and Seforno: It means that Mo'av became sick of their lives (because Yisrael were plundering them ? Seforno).
QUESTIONS ON RASHI
Rashi writes (in 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 But 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 va'Chomer, had Hashem not already forbidden Yisrael to fight 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 ? really, it was lifted only after Eglon harassed Yisrael; Yo'av answered [falsely] from a Torah verse, to avoid Chilul Hashem.