This is a great question, which is not easy to answer. A Google search reveals all sorts of answers, many of which amount to what folklorists would refer to as folk etymologies (specifically for word origins; but the concept applies here as well) - after-the-fact popular explanations of something common within one's own culture.

Some of what one finds hint at the issues that swirl behind the question of green skin. Many focus on the effects of torture on the skin of the accused during the witch trials - everything from bruising and gangrene. These are absurd on at least one level - I doubt any primary source documentation could be found to support this after-the-fact conjecturing. That said, green can be regarded as a color of putrefaction - even though it is also a color of vibrant life.

The most credible explanations point to the 1939 film "The Wizard of Oz" as the origin of this motif. This film was at the vanguard of Hollywood's experimentation with the color films, and no doubt the director and producers wanted to have the most effective use of color. How and why they decided on green skin for the witch is not something I can address. We needs the Mods to ask for a film historian amongst our ranks to address this. I see no evidence that the director was drawing on a popular concept that witches had green skin; instead it appears that this may have been an original, unique choice with intent of making the most effective use of the cinematic tool of color film. It is worth noting that Frank Baum does not describe the witch has having green skin in the book (1900). A color that would leap to mind would be blue - that was the way pre-conversion Scandinavians apparently viewed the skin of the Goddess Hel - blue being a color that a corpse would assume. But the producers of the film selected green.

Let's not forget that this was an extremely popular and influential film. It is hard to imagine a world were a film would not live beyond its initial run in the theaters, but in a world before television, HBO, Netflix, and HULU, films were made for the initial viewing and then were generally retired. Sometimes there was a second run or they could serve as the second feature in a "double feature", but that was it. Television provided a new means for showing films again, but that medium had to wait until after the war. "The Wizard of Oz" became one of the standards on television and it was shown every year so new generations of children could be introduced to it. Of course, early televisions were black and white, but eventually, the green skin of the witch became part of this annual tradition of watching "The Wizard of Oz."

It is one thing for the producers of a film to make the choice that witch's skin would be green; it is another for the folk to embrace it. For whatever reason, this became part of folk culture. The producers of the film clearly hit the right chord, and that is not an easy thing to do. The folk can be terribly fickle, embracing one thing and rejecting the next.