When water vapor condenses on a dry mirror, it does so as separate droplets, a process known as dropwise condensation.
The numerous drops effectively screen the mirror so that it appears opaque.
When you draw on the surface with your finger, the droplets coalesce into a thin film of transparent water, so the mirror becomes reflective again in these areas.
When the mirror warms up, or the air humidity falls, the droplets evaporate. The image disappears because the surrounding droplets no longer contrast with it.
The film of water evaporates more slowly than the droplets because of its lower surface area.
If it does not have time to completely evaporate, any condensation occurring soon afterward will be dropwise where there were droplets before, and in a film where some of the film remains. This latter process is known as filmwise condensation. The image then reappears on the glass.
If the mirror dries completely, the pattern should not normally reappear when further condensation occurs, though it might if the surface has been contaminated where you drew the image.
Drawing a finger across the mirror in making the image may leave traces of sweat on the surface which would, owing to its salt content, help to promote filmwise condensation.
Dropwise condensation is known to chemical engineers to be more efficient at transferring heat than filmwise condensation, but in practice it is much more difficult to promote, because as the droplets enlarge, they touch each other and coalesce, so the process tends to become filmwise.
On the other hand, dropwise condensation is easy to prevent.
Wiping the mirror with a cloth or a tissue wetted with a small amount of detergent such as shampoo leaves an invisible film on the surface.
This reduces the surface tension of the condensing droplets, causing them to flatten out and readily coalesce into a film. This is the basis of the anti-misting fluids which are used for treating spectacle lenses and car windshields.
When you draw an image in the condensation mist, you leave traces of finger grease, or, if you have just washed, grease plus shampoo or soap.
The film is transparent, so you don’t see it when the condensation clears. The next time water vapor condenses on the cold mirror, there is a difference in droplet size between condensation on clean glass and on contaminated glass.
In some cases, it is the contaminated glass that encourages droplet formation, and then you see the image as positive rather than negative.
But usually water-loving surfactants such as soap reduce the formation of droplets and generate a smoother, clear film of water, contrasting with the gray mist on the surrounding glass.