。
She was very nervous about her first flight.
Literal
She [topic-は] first [of-の] flight [about-に] very nervous had-become.
神経質になっていた combines the na-adjective 神経質 (nervous/neurotic) with になる (become) and ていた (past ongoing state) — she had become nervous and was still in that state. たいへん (very/greatly) intensifies the nervousness. 飛行 (flight) is the formal term for the act of flying; in casual speech, you'd more likely hear フライト or 飛行機に乗る (to ride an airplane).