I discussed the rise and fall of wigs during our April Fools day, and though it's "in character" it's all still accurate and referenced. The popularity comes down to how achievable the fashionable look is with the average persons head. I will expand upon the reason they fell from fashion, however. It's not just something we can attribute to the French Revolution and it's ideals, wigs were starting to fall out of fashion by the 1770s. Fashion overall changes a great deal during that decade, moving towards the more "English" sporting style. Trim is less intense and fluffy, even in the case of men's suits things like cuffs and pocket flaps are much reduced and streamlined. Women's hair, which was already more often their own rather than a wig, during the 1780s was very curly and natural. So we're already approaching a much more relaxed, sporting attire, some even inspired by the countryside and things like shepherdesses!
The final ending for wigs comes with the next fashion trend developing in the late 1780s- Classicism. Roman and Greek cultures were already very popular in the academic sense during the 18th century, but with the creation of the American Republic and the discussions occurring in France, Roman styles took hold in the fashion world as well. Laced sandals, high-waisted gowns, and hair styles "a la Titus" and "a la Grec". Men's hair is fashionable short (as is womens), so the need for wigs is negated. Keep in mind that hair pieces and other augmentations like cushions for shape have still been popularly used up to this day, it's just the full wig that has been less favorable with most.
Well, the answer to the first is Louis XIV. He started losing his hair and wore a wig, circa 1670, and soon all the men at court were doing the same. This is a very Versailles thing to happen, imitation of the king...when he was being "cut for stone", i.e. needing to have bladder stones removed, a number of nobles even volunteered to have the operation as well, even though they didn't need it. France being the fashion-setting country of Europe at the time, wigs were quickly adopted elsewhere. Maybe someone else will know precisely when and why wig-wearing stopped. It was somewhere around the time of the French Revolution, which leads me to suspect that it had to do with Rousseau's ideas about more "natural" life and a general trend for less artificial fashions, but that's only speculation.
You have noticed perhaps that English barristers and judges still wear wigs in court? Though I doubt any of them shave their heads anymore.