Why are communities that are more diverse usually more stable?

It is going to depend on what you consider stable. A diverse population would be more resistant to disease because of simple biology. The more sources for possible resistance the better the heterogeneous pool will be at resisting disease. You also have to take in to consideration things like the availability of modern medicine and the ability to be isolated during illness. Personally I think it has to do with the fact that many of the worlds more diverse population centers are also many of the worlds largest population centers which make them less prone to invasion on that basis alone.