To make an effort; to strive; to devote oneself to (doing something) with diligence.
彼は大学院の研究に励んでいる。
He is devoting himself to his graduate research.
毎日ピアノの練習に励めば、上手になりますよ。
If you practice the piano every day, you'll get better.
彼女は仕事に励んでいるが、趣味の時間も大切にしている。
She works hard at her job but also values time for her hobbies.