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She wouldn't let her child go out in the rain.
Literal
She [topic-は] child [target-に] rain [middle of-の中を] outside [to-へ] did-not-make-go.
Causative-negative: 行かせなかった is the negative past of the causative 行かせる ('make/let go'). With に marking the agent of the caused action and a negative form, the meaning is 'did not let X go' — the parent withholds permission rather than forcing nonaction. 雨の中を marks the path traversed (a less common but legitimate use of を with motion verbs). The causative-negative emphasizes 'did not allow,' framing the parent as the gatekeeper of the action.