To Write In
Transitive verb and Godan verb
かきこむ
急いで申込書に書き込んだ方がいいよ。あと十分で事務所は閉まっちゃうんだから。
You better hurry up and fill out an application. The office will close in ten minutes.
You write something on this paper but it is crowded. Why is it crowded? Because there are little boxes for you to write things in. You are filling out a form, which makes this verb to fill out or to write in.
The reading comes from 書く and 込む combined into one.