It's my belief that a large part of mathematical research, perhaps more than we would like to admit, comes down to finding clever elementary arguments.  This is particularly true in my own area (combinatorics and related fields) but it is true of many other fields as well.  Of course there's usually some machinery to be mastered, but at the end of the day, you're trying to come up with something new, and it's not so often that you're building some gigantic new machine out of whole cloth. Typically you're taking various known ideas and trying to figure out how to adapt them and put them together in a new way to prove something new.  When the pieces of the puzzle finally fall into place, I find the experience to be not unlike the process of solving an Olympiad problem.  The Olympiad training is useful for building a sense of confidence that something nontrivial can emerge with a bit of persistence and cleverness.  I find that some of my colleagues without this kind of problem-solving background will sometimes give up too quickly, because they are at a loss as to how to proceed when their usual box of tools doesn't apply.

A second way in which I find Olympiad training useful is that when I am confronted with a new and difficult problem that seems too hard to tackle directly, I can often find a way to invent a toy version of the problem whose solution may give some insight.  Experience with Olympiad problems has given me a sense of what a "bite-sized" problem looks like—subtle enough to be nontrivial, but simple enough to be tractable.  Interestingly, I often find that some of my colleagues who are better than I am at solving Olympiad problems can often solve my bite-sized problems when I can't; at the same time, I often seem to be better than those same colleagues at coming up with the bite-sized problems in the first place.  This may partly explain why some Olympiad stars don't become good mathematical researchers.  Research requires several skills, and those who only know how to solve tractable problems and don't know how to formulate them in the first place may not do so well at research.  But I think that experience with Olympiad problems can help with the formulation process as well.