。
Manjiro played a role in bringing these two countries closer together.
Literal
Manjiro [topic-は] these two-countries [object-を] each-other intimate [adverbial-に] make for [purpose-ために] one-role [object-を] fulfilled.
A reference to John Manjirō (中浜万次郎), a historical figure who helped open Japan-US relations in the mid-1800s. 一役果たす ('to fulfill one role,' 'to play a part') is an idiom for contributing to something. お互いに ('mutually,' 'each other') pairs with the reflexive/reciprocal sense of 親しくする ('make close'). ために sets up the purpose of the action.