Aramaic rose to prominence as a lingua franca in the Neo-Assyrian period. The Achaemenid empire, which adopted many aspects of Neo-Assyrian imperial administration and courtly organization and practices, likewise used Aramaic for administration and diplomacy. The relative ease with which one could master the Aramaic script made it particularly suitable for meeting the needs of a burgeoning bureaucracy.

There is always more to be said on the topic, but see my response to How exactly did Aramaic somehow replace Akkadian and become adopted by much of the Neo-Assyrian Empire? and the response by u/Daeres to Why did the Achaemenids make Aramaic the language of their empire?