Some trivia game shows
ask questions to one contestant at a time instead of pitting them against each other directly with a buzzer. For example, Alice will answer a set of questions, and then Bob will answer a set afterwards.
The problem with this is, if Alice and Bob are asked different questions, it doesn't feel fair, and if they are asked the same questions, then Bob will know the questions ahead of time when they're asked to Alice.
This is where the sound-proof booth comes in. Bob is placed in a large glass booth through which no sound can enter, while Alice answers the questions. Bob usually wears headphones as well, just in case. The host is careful to let us know the purpose of the booth every time it's used, even though it's usually pretty self-explanatory. An odd tendency is for the host to whisper this fact to us, seemingly simulating what it sounds like to the person in the sound proof booth (in which case it isn't very sound proof at all).
As to why they couldn't just put Bob backstage: a sound-proof booth is just cooler
The sound-proof booth has also been used in beauty pageants, including those in the Miss Universe organization, when the contestants are asked a final question before the judges decide the winner.