。
She's busy looking after the kids.
Literal
She [topic-は] children [genitive-の] care [with/by-で] is-busy.
The particle で here marks the cause or activity that someone is busy with — '[busy] with X.' This 'cause/means で' is one of で's most common functions outside its location reading. 忙しい takes the cause/topic of the busyness in this slot rather than the more obvious が or を, a particle pattern that's worth memorizing as a frame: 仕事で忙しい ('busy with work'), 子供で忙しい ('busy with the kids'), 試験勉強で忙しい ('busy with exam study').