It took years of dedication for the young musician to make his way onto the international stage.

make 在这里是动词，意思是"实现、达成"。 his 是代词 he 的所有格形式，表示"他的"。 way 的意思是"道路、途径"，引申为"成功之路"。 make one's way 是一个固定短语，意思是"（通过努力）取得成功/获得地位"，常用来描述通过长期努力逐步达到某个目标的过程。