Oh, really? I didn't think so because of your reply "An expression of conjecture". As Elizabeth-san explained, でしょう is added for politeness, similar to ですか?.
In this case, どうして is modifying 書いた, i.e., the order of words can be changed to 勉強しないのに、どうしてアートについて書いたのでしょう?. Thus, she was asking to explain the reason why you wrote; 「(勉強しないのに、)どうしてアートについて書いた」の?