Wigs were, up until the end of the 18th century, a huge part of the traditional aristocrat dress. Of course, the final few years of the 18th century were a pretty rough time for the traditional aristocracy. Of course, the French Revolution was largely a revolt against the aristocracy, and the new upper class systematically rejected huge parts of the old aristocrat lifestyle. The wig-wearing old aristocrats were largely driven out of Paris, stripped of their wealth, or killed during the reign of terror, and were replaced by some of the more entrepreneurial members of the third estate.

This new upper class, created because of the revolution, had largely come from relatively working class backgrounds, what would be referred to as 'Nouveau riche'. Many made huge fortunes controlling weapons during the revolution, and quickly found themselves in positions of power and influence. Like their third estate bretheren, they'd fought to be rid of the out-of-touch aristocracy, and so didn't take up some of the more extravagant lifestyle choices, one of those being wigs.

Dressing in exaggerated forms of working class clothing, like the infamous Pantaloon (trousers) and ties, they represented a shift from the Ancien Regime and pioneered what would become the modern day suit. The new form of dress was so revolutionary, its style would actually be referred to in France as 'L'incroyable' - the incredible, for quite a while.

Why did that catch on in the rest of the world? Paris had been a centre of culture and fashion for a long time before the revolution. The centralised and extremely powerful French nobility sucked the wealth out of the money the monarch made from his trading adventures and colonisation. This made for one of the most ridiculous and extravagant nobilities in all of Europe, and though this turned out to be their downfall, it enabled them to pioneer all kinds of crazy styles. Because they were so leading in these areas, gradually the rest of the worlds nobles turned to Paris to get fashion advice, and so when tights and wigs were swapped for trousers and ties in Paris, it didn't take long for it to be the norm from London to Vienna.

TL;DR - the French revolution gave the wig a bad name.

Source: Various lectures on the subject at Cambridge University.