It's a silly superstition among coaches who don't understand psychology.
Before you read the rest of my response, grab the nearest pen/pencil and put it up behind your ear. Seriously, do it.
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No, I mean it. Don't read ahead to see where I'm going with this, just do it.
Ok, now to continue...
The theory behind it is that feeling the band there is supposed to remind kids to keep their belly squeezed. This is bogus, because it's a continuous and unchanging stimulus, and as such the brain tunes it out after a minute or so. We do this with any stimulus that is constant and unchanging. For example, I bet before reading his sentance, you were completely unaware of the hum of your computer fans -- you can hear it when you're thinking about it, but otherwise you completely tune it out. I bet you don't feel your clothes on your body, or the hear the hum of the air conditioner or the refrigerator in the other room -- if you focus on those sounds, you can pick them out, but otherwise you reflexively tune them out.
The same is true of these waistbands. Unless they're tight enough to be painful, the gymnast will completely tune it out after the first minute or so, and it will therefore be completely inneffective as a reminder of anything.
Want one more example? Before reading this sentance, were you still aware of that pen sitting on your ear?