Lily doesn’t understand why the men from the gas station call the policeman “Shoe.” How would you explain the implications of this nickname to her?

I would probably be in the same boat as Lily. Early 20th Century, policeman or detectives were often referred to as gumshoes. which suggested they moved stealthily or cautiously. Their shoes were made of gum rubber, a substance that was quiet and allowed people to quietly. The name eventually came to be applied to detectives and it stuck.