People just sit at tables for an hour and try out solutions. Then they write it down and hand it to a judge.
You can read the details in the official rules, but basically you're allowed to have up to 3 of your own cubes (you must supply the cubes), paper, and pen. You may also have colored stickers to place on the cube if you find that helpful (it can be helpful for finding insertions). They give you a piece of paper with the scramble on it, along with spaces to write your solution. At the end of 60 minutes, you must be finished writing and hand the paper in to the judge. Then the judges try to find time before the end of the competition to do the tedious job of checking all the solutions to find out which ones are valid. This is often not a fun job (especially with the pressures of the rest of the competition to deal with at the same time), so please be kind and thoughtful to your judges.
What? :O Couldn't it just be manipulated in some way to find an easy solution?