How do the "proposed research" essays of those who win NSF Fellowships compare with those who don't win NSF Fellowships? I just looked at the research proposals of a few people who failed to win the NSF Fellowship, and they actually scare me a bit.

I know that the NSF cares about Broader Impact a lot, and that is where I might actually be strongest at. But the reviews look at intellectual merit first and foremost, and that's what scares me, since how can you convince the reviewers that your idea has more intellectual merit than a huge number of other very strong applicants?
As someone who reviews graduate applications, both for admissions and for fellowships, I can say that the number of strong applications is really large, and now in many cases is almost certainly larger than the number of fellowships being awarded in most of these competitions. The unfortunate side effect of this, which is obvious here, is that in most cases, you could have an application that is "strong enough" to merit receiving the fellowship, but not get it. Unfortunately, you don't get to do multiple stochastic realizations; you have to deal with the events the one time they unfold.

That said, how do you impress the reviewer? By having a clear sense of what your project is, and being able to talk about coherently and convincingly. The reviewers are looking for good ideas and good people to fund. You need to convince them that you're going to be a good researcher, and you have a good idea to "sell." If you don't think your research is all that amazing, how are you going to convince somebody else about that?