I was going to quote a few of the people above, but their were enough good points in different posts that it was difficult to put them all together.
The basic reason for warming up is to prepare yourself to perform an athletic or physcial activity. This preparation is not just limited to your body, however, it also involves the mind.
The best athletes in the world in all sports have a standard warmup routine that they go through every single time they prepare for their sport. Not just in competition, but also for practice. This has the effect of telling both your mind and your body "ok, it's time to get going."
It's important to develop a clear warmup that has a routine to it, from the order you do things (run, stretch, fence, etc...) to the order you stretch (I personally start at the top and work my way down). While you may not see results right away, in the long term you will be developing a signal that tells your mind to get prepared. As mentioned above, it will significantly improve your focus, which will lead to improved results and improved consistency.
For many people this routine can actually extend to well beyond the day of the tournament, sometimes going as far out as a week before. Mine personally starts the night before when I like to eat a relaxed dinner with friends, hang out in my hotel room and watch TV, maybe fix my weapons if they need it. The day of, I'm at the tournament an hour before the close of registration, and running laps 45 minutes before the close. Then I stretch, do some footwork (typically bout oriented), suit up and fence a few bouts concentrating on choosing actions and getting my distance, timing and tactics ready. By the time the pool starts I may have physically cooled down because of delays or other reasons, but mentally I am completely prepared for my bouts ahead.
I've discovered in Paris and at nationals last year, how detrimental it can be to my performance if I'm not able to go through that process, or if any portions are disrupted.
While Inq sees no relation between his warmup and his performance, in the long term it can be a tremendous boon that allows you more control over your results. If you suffer from nervousness, the routine can have a calming effect, if you struggle to be intense enough the routine can bring you up.
An effective routine may not spring into existence the first time you go to a competition. But over the course of a year, by charting your performances and analyzing why you were successful or not you can develop a system that will help you kick as much *** as you've been striving for all these years
edit: oooo, long post, sorry.