Why can you see your breath on a cold day?

Because you are making a small cloud. Air leaving your lungs is warm and saturated with water vapour; it meets cold air, cools below the point where it can hold that much vapour, and the excess condenses into droplets you can see. It is the same process as a cloud, a mist and the fog on a bathroom mirror.

The temperature it starts at depends on how humid the air already is. On a damp day breath is visible at higher temperatures because the surrounding air is already close to saturated; in very dry cold air it can be invisible several degrees below freezing.

The same physics produces contrails behind aircraft, where engine exhaust adds both water and particles for droplets to form on, and the reason you can sometimes see them for hours and sometimes not at all is the humidity of the air at that altitude. In extreme cold - below about minus 40 - exhaled moisture freezes directly into ice crystals, which Siberians call the whisper of the stars because it makes a faint rustle.