It suggests that the person is being childish or overly dramatic in their expression of unhappiness.
So, essentially, it means:
* To pout: To push your lips forward to show you are annoyed.
* To sulk: To be silent and bad-tempered because you are annoyed about something.
* To be in a bad mood and showing it visibly.
The tone is usually playful or slightly sarcastic, implying that the person's distress is a bit exaggerated or unnecessary.