This one confuses me too for the same reason.

"The birds are not inside, but outside" would be translated as "A madarak nem bent vannak, hanem kint." In that case, the adverbial of place is negated. But, we have no other clause here, so the negation is targeting the whole sentence. That is expressed by "(they) are not". However, there is no *"(ők) nem vannak" in Hungarian, but "(ők) nincsenek". "There are no birds inside." would be "Nincs bent madár." (Note that this is only in present tense, and only for 3rd person, both in singular and in plural.)

"There are no birds inside" can be translated to "Nincsenek bent madarak", but since we have the definite article "the", it cannot be translated like that, we need the "a" in Hungarian. In this case, it can be arbitrarily translated as

• " Nincsenek bent a madarak", or
• "A madarak nem bent vannak",

because we don't know, which one was emphasised in the English sentence.