If you do something arduously, you put a lot of hard work into it. When a runner trains arduously for a marathon, she practices in a dedicated, rigorous way.
A music student training to be a professional violinist has to practice arduously, and a swimmer fighting against a strong current must swim arduously to make it back to shore. In either case, there is difficult, focused work being done. The adverb arduously comes from its adjective form, arduous, from the Latin root arduus, literally "high or steep," and figuratively "difficult."