What did the nations hear that caused them to tremble?
Ramban #1: It refers to the Might of Hashem, that He had just displayed at the Yam-Suf - when they would hear about it.
Ramban #2: It refers to the havoc that Hashem had wrought in Egypt, and they were overcome with an ongoing fear that they would not suffer the same fate. 1
Hence Moshe prayed (in Pasuk 16), that that fear should overcome them to the extent that they will not dare leave their cities to fight Yisrael (Ramban).
Why did terror grip specifically the Pelishtim?
Rashi (citing the Mechilta): Because they killed the b'nei Efrayim who left Egypt early. 1
Rashbam: Because they (as well as the chieftains of Edom and the strong men of Mo'av, mentioned in the following Pasuk) were neighbors of Yisrael.