。
The plot of the story unfolds on an island.
Literal
Story [possessive-の] plot [topic-は] certain island [object-を] stage [location-に] unfolds.
筋 (すじ) here means 'plot' (of a story). ある島を舞台に means 'with a certain island as the stage' — 舞台 (stage/setting) is used metaphorically for where a story takes place. 展開する means 'to unfold/develop.'