Breathing in human body is done through lungs which exchange air. They are almost moist and warm. Now when we breathe, we inhale oxygen rich air and exhale carbon dioxide rich air.
However as the the lungs are warm and moist ,the exhaled air is warm and has more moisture in the form of water vapour.
In cold,when this warm and moist air comes in contact with the cold air of atmosphere ,the moisture in warm air condenses to form tiny water droplets. Thus, we see this tiny droplets as cloud or fog in our breathe.