I think the easiest way to get to this answer is the process of elimination.
「はい、そうです」 is out because the question is weighing yes/no answers equally but this answer is just confirming what the speaker said, which only makes sense when the speaker emphasized one over the other, such as 「質問はありませんね？」
「いいえ、あります」 is out because if the answer starts with "No", what needs to follow is a negative answer, such as 「いいえ、ありません」
I actually think 「はい、ありません」 is a rather unnatural answer, but it certainly is the least unnatural answer of the three. The key to understand why this answer makes sense is that 「はい」 here is an interjection, not a positive answer. But precisely because of that confusion, I'd never answer this question with 「はい、ありません」. I'da rather use 「いいえ、ありません」.