The mother nagged her son until he listened.

Literal

Mother [topic-は] son [to-に] noisily said-[sequential-て] made-listen

やかましく = 'noisily/naggingly' (adverb from やかましい). 言って聞かせた chains 言う + 聞かせる (causative) — she spoke and made him hear. The compound implies persistence until the message landed.