"What does it take to win a homebrew competition?" The simple answer is: an Excellent to Outstanding homebrew that is a classic example to the style without flaws and leaves a wonderful impression.
A lot of focus has been put on unreliable judging. I think that's a silly argument...can you get an incompetent judge? Absolutely. I see little point worrying about things that may or may not be the case.
Before entering a competition, you need to ask yourself one simple question. Can I handle someone calling my baby ugly? If you can't cope with that potential fact, then don't enter into a comp.
I entered my first comp after 2 years of homebrewing. My friends and family told me I was brewing good beer, and I thought I was brewing good beer, but I wanted an unbiased opinion on my beer, plus I'm a competitor and I wanted to see how my beer stacked-up against others.
I did pretty well that first comp and felt I was on the right path.
There is always a potential to get bad advice from a trusted source and Beer Judging is no different, so yeah, sometimes you have to weed through the info. However, I am one that firmly believes that taking critiques from judging comps is a great way to improve your beer. If you want to Win a beer comp, sometimes you have to go through a loss and find ways to get better.
For my personal case study, in my first comp, I ended up scoring a 33/50 to win a Silver Medal (which is a really weak win). I took what the judges said to heart, worked on the recipe, and 3 years later submitted the beer for judging. Scored a 45/50 and took Gold.
Now? I'm done competing!

I know I have the potential to make Outstanding beer. Plus, because I have run the gambit of having beers judged from Fair to Outstanding, I feel I'm a little bit more calibrated and have become more unbiased about my beer and a better homebrewer because of it.