両国間の文化交流が進むにしたがって、相互理解も一段と深まっていった。

、。

As cultural exchange between the two countries progressed, mutual understanding deepened further.

Literal

Both-countries-between [possessive-の] cultural-exchange [subject-が] advance [as-にしたがって], mutual-understanding [also-も] one-step-further deepened-went.

~にしたがって 'as X progresses / in accordance with X' is a formal temporal-proportional pattern: as one thing advances, another follows. 一段と 'even more, a step further' intensifies 深まる 'to deepen.' ~ていった uses the auxiliary ~ていく to mark a gradual change progressing away from a reference point — the understanding kept deepening over time.