The U.S. government attempted to open Japan to the outside world.
Literal
American government [topic-は] Japan [object-を] open-country cause-try did.
開国させようと試みた layers causative (させる) and volitional (しよう) with 試みる (to attempt): 'attempted to make Japan open.' 開国 is the historical term for Japan's 'opening' to foreign trade in the 1850s — a crucial moment in Japanese history. The causative marks the U.S. as the agent forcing Japan to open.