Short Answer: Gymnastics requires the athlete to be as unrestricted as possible when performing his/her routine. The less physical obstacles like height, air, and yes, even leotard material there is, the faster/more efficient the athlete can execute their skills.
I have to say, this question usually comes from non-gymnasts. And understandably so, if you are not involved in the sport, it could be confusing to see why are "uniforms" are different and "more revealing" that other sports.
But we actually are not that different! Swimming has athletic wear similar to ours. So does diving, cycling, track and field, wrestling, triathlon, dancing, acrobatics, the list goes on. So you are really asking about why do some sports (like gymnastics) have tight clothing?
The Science Behind It
To put it simply: the less drag an object has in the air, the more aerodynamically efficient it is. What does that mean? To be more aerodynamically efficient in sports means reducing the resistance that air exerts on an object or athlete moving through it. In sports, minimizing drag is crucial for achieving higher speeds and improved performance. So in gymnastics, that means to flip and twist faster, jump higher, bend longer. Having a tighter leotard is our way of reducing drag. Putting our hair in a tighter bun, and not leaving it down is a way to reduce drag.
The Safety Behind It
Imagine doing a back flip with loose clothes, not a good idea, right? Well, for gymnasts, they are almost always doing a skill that involves flipping, and twisting, leaping, bending, etc. We want to minimize the risk for injury PHYSICALLY by not having any physical barrier in the way and MENTALLY for the gymnast to be focusing on his/her routine rather than a possible wardrobe malfunction.
The History Behind It
Refer back to my post about why the leotard looks the way that it does today. Long story short: people realized that in order to be efficient to flip in the air, the coverage of the leotard must be minimal. The actual beginning of gymnastics had athletes wearing no clothes at all, because that was the most aerodynamically efficient! So let's be thankful that we now can have something to cover our bodies comfortably enough without sacrificing aerodynamics!
The Showcase Behind It
There is something very powerful about seeing the strength of a gymnast's body on them. Another reason leotards are tight is to showcase the athlete's strength, flexibility. It also makes it easier for the judge to pinpoint and appreciate the gymnasts' precision in a skill. It gives a spotlight to the hard work that's put in.