What exactly, does "Refa'im Yechashvu Af Heim ka'Anakim" mean?
Rashi: It means that the Eimim were considered Refa'im (like the giants, which is why they were called 'Eimim' - Ramban). 1
Ramban #1: It means that they were important 2 like the giants (in the era prior to the Flood).
The Refa'im appear a number of times in the Parshah (here, in Pasuk 20 and in 3:13). What is the connection between them?
Rashi: The first two were called "Eretz Refa'im" but were not the Refa'im that Hashem promised to Avraham; the third one, which Yisrael captured from Og Melech ha'Bashan, was. 1
Ramban: All three lands comprised the vast territory that Hashem promised to Avraham, only Hashem gave two of them to Amon and Mo'av, whereas that which the Refa'im still retained, over which Og Melech ha'Bashan reigned, Yisrael conquered. 2
This is what the Ramban himself writes on Pasuk 23 - based on the fact that, when listing the ten nations in Bereishis 15:19-21, the Torah mentions the Refa'im but not the Chivi - 'Our Rebbe'im all agree that Yisrael did not inherit the Keini, Kenizi and Kadmoni (because they were given to Edom, Amon and Mo'av' - See Rashi there, Pasuk 18 - 'but that they are destined to inherit it in the future, and it is to them that the Torah is referring in 18:9. Nor do we find that Yisrael inherited their land in Seifer Yehoshu'a. That being the case, the Refa'im which Hashem gave to Avraham are synonymous with the Chivi, the sixth son of Cana'an, whom they nicknamed 'Refa'im' in the days of Avraham.' (This explanation, which substitutes the original 'Chivi' for the later 'Refa'im', seems to conform to that of Rashi (Refer to 2:20:2:1, 2:20:2:2*). However, when he writes (in DH 'v'Hinei ha'Kasuv') that the current Pesukim are describing what happened to the territory of the Chivi (alias the Refa'im, the Eimim and the Chori), that part went to Mo'av, part to Eisav and part to Amon, which were all forbidden to Yisrael, and that the remainder which was taken by the Kaftorim, was now permitted to Yisrael, he seems to have reverted to his original explanation, according to which it is not clear what happened to the Keini, Kenizi and Kadmoni.
Why were the 'Eimim' and the 'Refa'im' called by these names?
Rashi: Refa'im - because whoever saw them 'Yadav Misrap'os' (became weak with fright). The Mo'avim called them Emim, because their Eimah (dread) spread over whoever saw them.