Believe it or not, there's nothing magical about seeing your breath when it's cold outside. It's just science at work.
You may already know that when you breathe in, your body takes in oxygen from the air. When you breathe out, your lungs expel carbon dioxide back into the air. But the breath you breathe out contains more than just carbon dioxide.
When you exhale (breathe out), your breath also contains moisture. Because your mouth and lungs are moist, each breath you exhale contains a little bit of water in the form of water vapor (the gas form of water).
For water to stay a gas in the form of water vapor, it needs enough energy to keep its molecules moving. Inside your lungs where it's nice and warm, this isn't a problem.
When you exhale and it's cold outside, though, the water vapor in your breath loses its energy quickly. Rather than continuing to move freely, the molecules begin to pack themselves closely together. As they do so, they slow down and begin to change into either liquid or solid forms of water.
This scientific process is called condensation. When you exhale when it's cold outside, the water vapor in your breath condenses into lots of tiny droplets of liquid water and ice (solid water) that you can see in the air as a cloud, similar to fog.
When it's warm out, though, the invisible water vapor gas stays invisible, because the warm air provides energy that allows the water vapor to remain a gas. As temperatures drop, it's more likely that you'll be able to see your breath.
There's no exact temperature at which condensation will occur. Many environmental factors other than temperature can play a role in condensation, including relative humidity (the amount of moisture in the air). When it falls below 45° F, though, you can usually expect to be able to see your breath.