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She slammed on the brakes when she saw a child dart out into the road.
Literal
She [topic-は] child [subject-が] road [onto-に] dart-out [nominalizer-の] [object-を] saw [then-て] strongly brake [object-を] applied.
A dramatic little driving scene built on the perception frame ~のを見て ('saw X happening, and...'). 飛び出す is a vivid compound verb literally 'fly out, leap out' — used for sudden, often dangerous, emergence into a space: a child into the street, a deer into the road, a fish from a pond. ブレーキをかける ('apply the brakes') is the standard collocation, with かける ('apply, hang') taking ブレーキ as object. 強く ('strongly') is the adverbial form of the i-adjective 強い — note い becomes く for adverbs.