I don't understand why anyone would want to work on the Riemann Hypothesis. It doesn't seem to be a problem that matters.

Could the fact that it doesn't seem to have many practical applications is what attracts certain people towards it? It doesn't have practical applications -> it's "purer" math. You're not trying to solve the problem for some external reason or using the math as a tool, you're trying to solve it for its own sake. I remember reading studies that mathematicians are on average more religious than scientists in general and I've also gotten the impression that some mathematicians relate to math a bit like it's religion. There is also this concept: http://en.wikipedia.org/wiki/Mathematical_beauty

It could be that some are just trying to impress others but I don't think it's always that simple.

And to my knowledge, there is some application for almost all the math that's been developed. Of course, if you optimized purely for applications, you might get better results.