To thoroughly understand and master reading a text or book — to digest its content so you can use or interpret it effectively (often used for difficult texts).
留学前に専門書を日本語で読みこなせるように練習している。
I'm practicing so I can read specialized books in Japanese before studying abroad.
その評論家は難解な作品でも読みこなして的確な批評を書くことで知られている。
That critic is known for being able to digest even difficult works and write accurate critiques.
初心者でも慣れれば新聞を読みこなせるようになる。
Even beginners, with practice, will be able to read newspapers competently.