The tone of a story could mean the vibe or mood it sets for the reader. You might as well say it's the music that is playing quietly in a scene of movie which kind helps to establish atmosphere and feeling. It consists of the author's choice of words, the setting, and the characters' actions. It could be humorous, suspenseful, tragic, romantic, dramatic etc.
For example, take 'The Hunger Games'. I know what you mean by that: really serious and heavy going, isn't it. Well, that was just the author using a serious or partly somber tone to fit in with his plot Contrarily, a book like 'Diary of a Wimpy Kid' uses a light, easy humor tone to go with its content. Tone can have a powerful effect on your reading of the story itself!