When the water vapor, that you breath out comes in contact with the cold air outside, it forms tiny water and ice droplets, which altogether appear as a cloud of breath. The reason the water droplets clump together is due to the lost of energy. When the warmer air comes in contact with the colder air it loses energy. As a result molecules slow down and lump together.
Another reason you see your breath is due to the rapid change in dew point that the moisture you exhale creates in the surrounding air.
Keep in mind that every breath we take in. we also exhale gases. We all know that the main gas we breath out is carbon dioxide but we also breath out a little bit of oxygen as well as water vapor too.
Your breath is not visible on hot days, because the warm air supplies enough energy to the water vapor to keep it in a gaseous state and warm air can hold more moisture.