A hot shower adds a great deal of water vapour to the air. The mirror is usually cooler than that warm, moist air. Air touching the glass cools and can no longer keep as much water in the vapour state. When the local temperature falls below the dew point, some water condenses on the glass. The “fog” is not white smoke suspended in the room, but a layer of extremely small droplets.
A dry, smooth mirror reflects light in an orderly enough way to form a clear image. Once droplets cover it, their curved air–water and water–glass boundaries refract and scatter incoming light in many directions. Fine detail and contrast are lost, so the reflection looks pale and blurred. Wiping a dry path immediately restores the image there; warming the mirror or waiting for the droplets to evaporate clears the rest.
This is also why heating the mirror, increasing ventilation or lowering room humidity can reduce fogging. These measures do not push “fog” away; they make condensation at the surface less likely. Anti-fog coatings take another route, encouraging water to spread into a more even film rather than collecting as many separate droplets, which reduces strong scattering. Their effect depends on glass temperature, humidity and the surface produced by the coating.
https://www.britannica.com/science/condensation-physics
https://www.weather.gov/jetstream/clouds
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