。
She's busy with her English studies.
Literal
She [topic-は] English [genitive-の] study [with-で] busy.
で here marks the cause or source of being busy — 'busy because of / busy with X.' This is one of で's broad functions: alongside marking instruments (ペンで書く), locations of action (公園で遊ぶ), and means (バスで行く), it can mark the reason or source of a state, especially with adjectives like 忙しい ('busy'), 疲れた ('tired'), or 嬉しい ('happy'). The plain 忙しい is an i-adjective in casual present-tense form. A more formal phrasing might be ~のために忙しい, but the simpler で is far more common in conversation.