You have good answers above, but one thing I didn't see is a mention that how much something will dissolve in hot water and how fast it dissolves in hot water are two different things. They usually go together, but not always.

Most, but not all, things will dissolve more in hot water than cold. It's tough to explain why that is in li5, but has to do with the universe liking it when things can move around more (we call this higher entropy), and that's what happens when most things dissolve in water- they get to float around and mix. When things are hotter the universe apparently likes it even more when things can move around more, so more stuff dissolves. There are some things that dissolve even less in hot water because the water has to organize or be shaped a certain way to get the thing to dissolve (the water is inconvenienced and can't move around as much), but that doesn't happen too often. I don't know why the universe likes it when things move around more, and likes it even more at higher temperature, but that's what a math equation called the Gibbs Free Energy equation tells us, and it seems to work.

But with washing hands what may be more important is that anything that's going to happen in water will happen faster when it's hot. I think this is always true- apparently water even freezes faster when it's warmer, as long as it's still cold enough to freeze. As others have explained above this is because the water (is "water molecules" acceptable for ELI5?) is moving around faster when it's hot, and a hotter temperature is very much like having a fast forward setting on, and cold water like being in slow motion. Your hands would probably wash ok in cold water, but it'd take longer (unless youre using one of those soaps that's made to work best in colder or room temperature water).

Another effect is that hot water is a little bit thinner and sticks to itself less than cold water, but that's probably not the main reason it cleans better. It should help dirt and germs get "wet" and start to loosen up a little bit faster though.