My mother said to me, 'Don't make noise.'

Literal

Mother [topic-は] me [to-に] 'make-noise-[must-not-てはいけません]' [quotative-と] said

「騒いではいけません」is direct speech — a polite prohibition within quotes. ~てはいけません = polite 'must not.' The mother is quoted verbatim, preserving her polite phrasing.