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She's tied up with raising three children.
Literal
She [topic-は] three [genitive-の] child-rearing [target-に] tied-up is.
かかりきり ('completely tied up with, fully occupied by') is a vivid noun-adjective built from かかる ('hang on, depend on') in its 'be occupied with' sense + the suffix ~きり ('only, exclusively'). Common in family and work contexts where one task absorbs all attention. 子育て ('child-rearing') is a compact noun built from 子 ('child') + 育て (the masu-stem of 育てる 'raise'). The に marks the absorbing task. The number 三人 modifies 子供 indirectly via の: 三人の子育て ('the rearing of three children').