The reason you can see your breath when it's cold outside is because of the temperature of the outside air and the temperature inside your lungs.
The temperature inside your lungs is warmer and when the cold air hits the warm air it produces condensation and that is why when you breath out you can see your breath.
When you breath out the water vapor from your breath condenses into tiny water droplets of water and ice that you can see in the air as a cloud just like fog.