This is not a normal English grammar structure.
The rules of Jeopardy require that contestants answer the clue in the form of a question.
I tend to think of it this way... it's a backwards game show. Most game shows require that the host ask a question:
Who was the first president of the United States of America?
And contestants provide the answer:
George Washington.
With Jeopardy, it's the other way around. The host is answering the question while the contestants are asking it...
So, if the clue is:
This man was the first president of the United States of America.
The rules require the response be
Who is George Washington?
If they fail to do this, even if they're technically correct, they lose points.
So, if a contestant answered:
George Washington.
They would be wrong by the rules of the game.
As to their hesitant tone, sometimes they're not completely sure of the answer, so they're guessing.
Neither "What are you after?" nor "Which one are you after?" would make sense here.
To be after something means to be trying to get it. This suggests that there is some effort required to obtain what you are looking for. For example:
What type of job are you after?
Usually, a menu item in a restaurant is not a difficult thing to obtain.
Here you would normally say "What would you like?" If a list of options the customer must choose from has already been mentioned, then you might instead use "which one." For example:
We have turkey, ham and salami sandwiches. Which one would you like? - I'll have turkey, please.
Best Answer
It really makes no difference to the meaning of the question which of the two you use, in today's common usage. Using 'which' sounds slightly more proper to me. I imagine that if the Queen of England were asking the question, that's how she would ask it. Needless to say, most of us don't speak like the Queen.
If you want to sound more natural you would probably say
or
The prepositions are at the ends of these two sentences, which we are often told is incorrect. Still, most English speakers will say it that way in casual conversation.