Translation:The singer affirms that I sing better than her.
Actually "better than she" is correct and grammatical but I doubt DL would accept it.
Why the 'người' here? I don't recall us using it for other professions or jobs (say, we weren't told to say 'người nhà khoa học' or 'người bác sĩ') in previous lessons.
"hay" is often used for singing or a talent. the other concern is one of correct English. "The singer affirms that I sing better than she (does.) "her" is incorrect english.
Duo is committed to "conversatiional," technically, incorrect usages. That is Duo's choice. The valid objection is that Duo often rejects correct and proper English.
Both “she” and “her” sound right to me, though one of them may be more correct than the other.