This goes back to Louis XIV and his minister, Colbert. In the mid-seventeenth century, the south of France - historically a textile-producing region - was economically devastated following the ever-roiling civil conflict between Catholicism and Protestantism, as there were a number of Huguenot strongholds there. Colbert had the textile factories brought back and instituted a system of strict bureaucratic controls, which both ensured high quality products and employed many more people. At the other end, regulations were put in place for aristocrats requiring them to appear at court in a constantly rotating succession of new clothes, which encouraged them to spend outrageous sums on their wardrobes. French industrialists imported fabrics (particularly silks and wools) from highly-regarded manufacturers (particularly in Italy) to work at imitating them to meet the needs of the domestic market. Dyers were forced to adhere to ingredients that provided the best - rather than necessarily the cheapest, or the okayest - coloring in the final product. As a result of the quality controls, French wares were very competitive in foreign markets, and because French weavers, dyers, lacemakers, tailors, etc. had a steady influx of income, the luxury industries boomed and could reinvest profits in manufacturing: success begetting success.

The important thing to keep in mind here is that while the concept of the cut/construction of clothes being important does date back to the middle ages, fashionable cuts did not change very quickly, and when they did change, the shift was often fairly subtle. (Particularly in court dress, which was regulated, as mentioned above.) Quality of material was a huge part of the overall quality and cost of an outfit, something that only stopped being true very recently. As a result, having a reputation for producing and wearing dazzling silks and laces at this point would therefore help to build a reputation for fashionability. That being said, at this time there was beginning to be a radical shift in fashionable women's dress construction: the introduction of the manteau (called a mantua in English), and its shift from a very loose kind of housedress into a formal everyday dress. Affluent women had been wearing a gown made of a separate, boned bodice and a petticoat; the mantua was a bodice combined with a skirt, opening down the front. These featured in some of the first commercial fashion plates, an art form very much associated with France as well!

Just as the mantua appears to have come to English fashion from France, the later sacque - a mantua with a flowing, pleated back - was also particularly associated with France. The English name itself shows French influence, although "sack" was a not-unheard-of spelling; in France, it was called the robe à la française, "gown in the French style", in contrast to the supposed dowdy Englishwoman still wearing a fully fitted mantua (which is why the robe à l'anglaise was named that way when it came back to high fashion in later decades). Sacques were hugely popular among the upper and upper middle classes from roughly 1740 to the mid-1770s, at which point a more casual and "English" ideal was taking hold. The fitted gown became what I'd call ordinary-fashionable, while looser gowns whose names were anglicized only the barest amount - the robe à la polonaise or polonese, the robe à la turque, the lévite, the chemise gown, etc. - were high fashion.

Why did these changes in dress construction catch on in Britain? Largely because they were novel, but also because they were French. As each fashionable thing was picked up from France in the late seventeenth and eighteenth centuries because it was seen as inherently attractive and desirable, it added to the cultural cachet of French fashion in the anglosphere. Certainly by 1800, using untranslated French terms and asserting that various styles had been imported from Paris were effective marketing tools. The early nineteenth century saw an increase in international name recognition for specific French dressmakers, culminating in the rise of Parisian couturiers with very large operations in the 1860s, such as Worth, Félix, and Pingat, whose capacity for advertising and attracting large audiences from Britain, America, and elsewhere were obviously bolstered by the many changes to business, travel, and manufacturing that were resulting from the Industrial Revolution.

(By that time, Paris was indeed the center of fashion for all westerners, but I simply can't tell you if the deliberate Parisian connection was as strong outside the anglosphere in the eighteenth century. Most sources on this in English focus on anglo reception of French fashions or the French industry itself. Language barriers! But the fashion plates I've seen from the late eighteenth century from Germany have indeed appeared to be following the new French gown types.)