My mother asked the babysitter not to take her eyes off the children.

Literal

Mother [topic-は] babysitter [to-に] children [from-から] eyes [object-を] not-let-go [indirect-command-ようにと] asked

目を離さないようにと頼んだ = 'asked (her) not to let her eyes wander from (the children).' 目を離す = 'to take one's eyes off' — 離す = to separate. から marks the source of attention.