。
She knows how to handle children.
Literal
She [topic-は] child [genitive-の] handling [object-を] knows-thoroughly.
心得ている is the resulting-state ~ている form of 心得る ('to understand, be well-versed in, have a thorough grasp of') — like other state verbs (知っている, 結婚している), 心得る in the bare present doesn't quite work; the ~ている is required to convey 'is in the state of knowing.' 扱い is the noun form of 扱う ('handle, deal with'); attached to 子供 by の it forms 'the handling of children.' 心得る is decisive and slightly formal — stronger than 知る or 分かる, implying mastery one can rely on.