Gymnastics requires extreme strength to weight ratios. Gymnasts are generaly short because of the physics governing fulcrums. The longer a boom or crain is the more weight has to be on the short side as a counter ballance. Hold a broom or pole out horizontally (parallel to the ground) is much harder when the pole is longer and eventually impossible if it were long enougn. somebody Archemedes (I think) once said "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world." In this case he would obviously be on the long end of the lever but it demonstrates the principal. I myself can probably curl maybe 150 pounds but if my forearm was say 4 inches long (a horrible thought) I could probably curl more like 600 pounds.
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They are usually thin because competative gymnasts train very hard. Most high level gymnast train from 16-24 hours a week. This of coarse keeps them in good shape. There are also plenty of full figured gymnasts in the world as well, but their intense training is the main reason you see so many thin gymnasts.
You need a high muscle to weight ratio. It's not to say you can't do gymnastics taller, it's just easier, so the best of the best are smaller. Like how monkey bars were easy as a kid, but difficult past 12.
Shorter gymnasts complete things that most of us on the taller side can't--for instance, bars or trampoline and floor. They have more room to run and more ability to do more flips since their legs are shorter. When you get above 5'4" or so, your strides are lengthy, which that takes away from the element of floor. On vault there is only so much space from the vault to the floor, so you want as much as you can possibly fit in that very short amount of time. Same with beam and bars.