"After him!" is short for "Go after him!" which means "to chase".
Tracker Tom is a private investigator (P.I.) who is trying to capture a robber or thief known as The Face. So, at the end of each page are phrases that mean "to run after or chase" the criminal before he gets away.
This writing style encaptivates the reader to feel like they are part of the story. Small children feel like they are the ones going after The Face. It makes the story "come alive" in a sense and is enjoyable for readers of all ages. Let me know if you have any other questions. I am happy to help!
Thanks for reading!