。
She wrote down her opinion on a piece of paper.
Literal
She [topic-は] her own opinion [object-を] paper [on-に] wrote-down.
書きつける (書く + 付ける, 'to write + to attach') is a compound verb meaning 'to write down, jot down, make a note of.' The 付ける suffix adds the sense of 'fixing in place' — emphasizing that the writing is meant to stick (recorded, not just spoken). 紙に書く would be the simpler version ('write on paper'); 書きつける adds a touch of deliberate or quick recording.