When you breathe in, a muscle under your chest, called your diaphragm, moves down and your ribs move out. This makes the space bigger and so you get lower air pressure in your lungs. Air now rushes in from outside. When you breathe out your diaphragm moves up and your ribs move back in, and the air gets pushed out
The model works in a similar way: When you pull down on the rubber, the space inside the bottle gets bigger and the air spreads out. You now have lower pressure inside the bottle, so the higher pressure outside pushes air in; the balloon is blocking the way, so it takes in the air. (This is like breathing in). When you push in the rubber the opposite happens – the air inside the bottle gets squashed up (higher pressure now) and this higher pressure pushes air out of the balloon. (This is like breathing out).