The answer is Hyperbole.
The Hyperbole is a figure of speech that shows exaggeration to show emphasis or to make a point.
This excerpt shows this in phrases like:
"I told her I hadn't a cent in the world"
"I talked low, and told her the whole story right from the start and nearly killed her laughing."
"What in the nation..."
"I never saw anything like it."
"I never saw a painful story"
All of these sentences in the excerpt demonstrate a type of exaggeration from the writer to add emphasis to them. Words like "never, in the world, killed her laughing", etc. aren't literal. They are used just to add this extra importance to each sentence and show contrast or attract reader's attention.