The text seems to say that when Moshe's hands were raised The Jews were victorious against Amalek and that when his hands were down thy suffered a loss. The Mishna asks "Is it possible that the hands of Moshe could make or break war? Rather, the Mishna states, when the Jewish People raised their eyes toward heaven they won and when they didn't then they lost." As cited in the Maayna Shel Torah (although I couldn't find it in the original work) the Sfat Emet says that it sounds like there's a connection between Moshe's hands being up and the people looking to G-d. Why then would he ever lower his hands? The Sfat Emet reads the Mishna to mean that when they looked to Heaven Moshe's hands got strength and when they didn't then Moshe's hands weakened.
The Sfat Emet's approach is strikingly powerful. He's saying that the leader needs to get strength from the people and then he feeds that strength back to the people. But if the leader's not getting from the people then it becomes harder, perhaps impossible. to keep his strength up, to keep giving. Thus Moshe could only lift up his hands to strengthen them if they were looking to G-d and inspiring him. It seems to me that this fits with why Moshe's hands were lifted by other leaders. He needed someone that was "into it" to give him support.
I don't hear this idea articulated often, but is ubiquitous in Jewish education. A friend of mine told me about being in a super class in high school. And the teacher shared a lesson with them that they ate up enthusiastically. The next day they asked her how it went over with the other class she taught it to. And she said, "They took notes," meaning that they didn't get it. This is a regular reality for teachers. It's not uncommon to have some students who thrive on what you teach and inspire you to work on your serve and others who don't even care about which side of the racket is up and that can make you less into it too. Teachers, like tennis instructors get recharged by students they can volley with.
This idea applies to all forms of leadership. It's hard when you give but don't get a response that reinvigorates you. This is why if you have any positive feedback for rabbis or teachers or lay leaders I urge you to tell them. In the end it's the community that wins or loses based on the strengthening of the hands that they provide or don't provide. If you're an looking to heaven along with them, they may know it and if they do it surely gives them strength. But sometimes a teacher or a preacher can't tell that they're being heard and they can start to lose their strength. They need to hear not just comments like "Nice speech," but things like "I was very moved by what you said about the need to learn with our children and to tell them stories, what book do you recommend." I actually heard someone say the latter to a rabbi and saw it give him strength.
I first heard of this Mishna in Rosh HaShana years ago in BMT. Rav Moshe Horowitz told us the idea and then led us in song, "Elaaaah Bizmaan..." The traditional explanation of this Mishna, the idea that our spiritual vision is key to our success, impressed me and has stayed with me. The Sfat Emet adds another dimension, that our attitude strengthens or weakens leaders and vice versa in a perpetual cycle. May we be blessed to look toward heaven in a way that leads to our success. May we be blessed to strengthen our leaders and in turn be strengthened by them.