The phrase is a metaphor comparing a narcissist who loses attention to something like a balloon losing air. The phrase means that a person goes from being excited and full of energy to feeling sad and tired. It also plays on the idea of having one's chest puffed out with pride and boastfullness and then, once the attention goes away, the narcissist exhales and their chest literally deflates as the air (and by extension the prideful boasting) goes away.