。
She sang softly and lulled the child to sleep.
Literal
She [topic-は] quiet-voice [with-で] singing, child [object-を] put-to-sleep.
寝かしつける is a compound verb built from 寝かす (to put someone to sleep — the causative of 寝る) and つける, which as a verb suffix conveys doing something with persistence or skill. Together it specifically means lulling or settling a child to sleep, a meaning neither part carries alone. 小声で uses the instrumental で to describe the manner of singing — literally 'with a quiet voice.' The whole scene is a natural domestic snapshot of a parent's nightly routine.