To Rush Into
intransitive verb, godan verb
駆ける is "to gallop" and 込む is "to be crowded." If you gallop into someone and crowd them, you are going to rush into them.
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.