Why is the word "turned" thrown in here? "Are you 21?" makes sense, "you are 21?" also makes sense, even though less people would ask that way. But the "correct" answer seems like a bit of a jump.
"Are you 21?" would be better translated as "Тебе 21?". Here we have "тебе есть 21?" which is slightly different. It means that the speaker doesn't really enquire about the addressee's exact age but rather wants to know if they are old enough for something that requires to be at least 21.
The Russian sentence is asking if the person is at least 21 (literally have they got 21 years).
The English idiom is to ask if the person has had that birthday (turned 21), which will necessarily be in the past if it has occurred. In English aging is a process that a person does, so being a particular age means you've done enough aging in the past to qualify.
Here it means to recently have become 21 years old. "turning 21" = "recently had your 21st birthday" or "recently become 21 years old".
Usually, "turning" an age is used to refer to important ages. For example, in the US in many States, teenagers can obtain a driver's license when they "turn 16". People can register to vote when they "turn 18". People can drink legally when they "turn 21". People qualify for government retirement benefits when they "turn 65".
Although you can"turn [other ages]", the phrase does not have the same impact or significance. Age 30 in the US has some significance - like leaving your youth behind you. Birthdays on decades are more significant than others.
And here I thought they were playing blackjack. "Why yes, I do have 21. Blackjack!"