。
Who was it that blurted out the secret?
Literal
Secret [object-を] talked [regret-てしまった] [nominalizer-の] [topic-は] who [copula-だ]?
しゃべってしまった uses ~てしまう to express regret or unwanted completion — the secret was blurted out, and that's a problem. The ~のは...だ cleft structure focuses on identifying who was responsible.