Chazal tell us that the attack of Amalek happened because "rafu Y'DEIHEM min haTorah," because Klal Yisrael neglected their Torah learning. Why the stress on "y'deihem?" The Maharasham in Techeiles Mordechai quotes the Chazal that "Torah mateshes kocho shel adam." That doesn't necessarily mean that a person who learns Torah is physically weak. The Torah warns us that a person should not think his success comes from "kochi v'otzem YADI," his own strength and ability. It's this koach -- the belief in the power of "kochi v'otzem YADI" -- which Torah weakens. Learning Torah humbles a person; the more one learns the more one recognizes the vastness and depths of the dvar Hashem and just how limited our own ability and insight is. Because Klal Yisrael's dedication to learning fell a notch, "min haTorah," because of that loss, "rafu y'deihem," their perspective on the limits of the "kochi v'otzem YADI" was tainted.
The Mahrasham does not say it, but I think based on his approach we can understand why Klal Yisrael had to fight Amalek as opposed to being miraculously rescued as had happened at Yam Suf. Since they invested trust in their kochi v'otzem yadi, Hashem responded in kind and let them use their own power to carry on the battle.
Moshe's challenge was to shift their perspective. "V'haya ka'asher yarim YADO..." Moshe lifted up his hands and directed people's gaze to shamayim. There is no power in "otzem yadi" -- our hands only have power when we connect to Hashem.
"Va'yehi YADAV emunah ad bo ha'shemesh" Ibn Ezra interprets emunah here either has steadfast or like the word "omain" (e,g. "Va'yehi omein es Hadasah"), to train, to nurture. Our hands need to be trained to connect to Hashem.
"YAD al keis K-h" -- the war with Amalek is won when we connect "otzem yadi," our ability, to the kisei of Hashem, the true source of power.