To Rush Into
intransitive verb, godan verb
駆ける is "to gallop" and the kanji 込 means into. To gallop into a place is to rush into it. You can’t gallop slowly, after all. This can also mean to seek refuge, as in hurrying to a place for safety and support.
かけこむ
The readings come from 駆ける and 込む.
終電に駆け込み乗車をしようとしましたが、目の前でドアが閉まりました。
I was rushing onto the last train, but the door closed before my eyes.