Transverse chromatic aberration: The focal length a measurement, lens to image plane, when imaging a far distant subject. The camera lens is a converging lens. Light from an object at infinity enters the lens as parallel rays. The lens bends these inward. The path of the image forming rays trace out a cone. The camera is focused. This adjusts the distance, lens to image. The image will be in focus when the apex of the cone just kisses the imaging chip (film).
Now all lenses suffer from 7 aberrations. These are mitigated by the optician by constructing the lens using several different shaped lenses, some dense glass, some less dense. Chromatic aberration is mitigated by combining a positive power lens with a weak negative. They have opposite chromatic aberrations so they almost but not quite cancel each other.
Transverse chromatic aberration is also called chromatic variation of focal length. What happens is: The distance lens to image is a function of light frequency. Violet comes to a focus near the lens while red’s focus is further downstream. All other colors such as green, yellow, and orange come to a focus in an intermediate position. The further the image forms from the lens, the bigger it is. Thus the red image is largest, violet the smallest. What we see is a color fringing, a rainbow of color surrounding the boundaries of each object being imaged. Again we can mediate but not eliminate with transparent lenses. The mirror lens, with a first surface mirror with slivering on the surface shows no chromatic aberrations as the light does not transvers the lens.
As to how software can mitigate. The rainbow pattern surrounding the subject is identifiable and thus the software can mitigate. The software looks at adjacent pixels and interpolates between. In a way the results are fabricated as this is just a best possible guess as to the colors near the periphery of objects.