It was at that moment that my father's nagging really hit home.
Literal
Father's scolding [subject-が] my chest [in-に] deeply struck [nominalizer-の] [topic-は] that-time was.
のは~だった is a cleft sentence focusing 'that moment.' 小言 means 'nagging/scolding.' しみじみとこたえた means 'struck deeply / hit home' — 胸にこたえる is an idiom for something emotionally affecting the heart/chest. しみじみと is a mimetic adverb for deep, quiet feeling.