"Travaille-t-elle dans cette ville ?"
Translation:Does she work in this city?
Nothing. Its purpose is to render the speech easier, since there are two vowel sounds one after the other (the last 'e' from travaille and the first one from elle). It's the exact same reason why, in English, we say an instead of a in front of words starting with vowel sounds (like an apple).
I thought "She works in this city?" would be correct. How would you say this instead of "Does she work in this city?"
I think you cannot really make these kinds of grammatical parallels between languages. Both mean the same in English. In French you could say either the sentence proposed or Est-ce qu'elle travaille dans cette ville ? or even Elle travaille dans cette ville ? (with the pronunciation different, just like in English, to mark it clearly as an interrogation when speaking). But you can't really say one way of phrasing in in one language corresponds exactly to one other way in another language (but maybe for the affirmative sentence pronounced interrogatively).
Did anybody else make the mistake of submitting "Travaillent-elles dans cette ville"? They sound exactly the same, don't they?
They do sound the same, so if you had the exercise spoken (and had to write it down), I'd suggest you repost it to Duo (with the "my answer should have been accepted" report button).