絮 originally means 'cotton wadding / fluff' that piles up. As a verb it took on the meaning 'to repeat at length' (think how fluff just keeps coming). Pair it with 嘴 ('mouth') and you get 'fluffy-mouthed' - chattering on and on.
register
Colloquial and slightly negative, implying the talker is wearing the listener out. 唠叨 is a near-synonym used in the same kinds of family contexts.
