Like most stereotypes, bears some relationship to reality (namely, what Americans think the French think of Americans) while being mostly wrong (actual French thoughts on Americans are more along the lines of "constantly-grinning workaholics who are chronically stiffed on vacation; how and why do they put up with that?"). One possible root in reality is because French etiquette is more formal and polite than American manners, which are more laidback, and as a result, French people might sometimes be perceived by Americans as arrogant and uptight. French humor does tend to be dark, but France itself (and fellow French) comes in for just as much ribbing as any other target.
Language might be another source of Values Dissonance here. After all, the French language contains many excessively nasal "ew", "uh", and "eur" sounds, which to the English-speaking ear tend to suggest disgust or snobbishness. The fact that most Americans have historically hailed from the eastern part of North America, where those sounds (inasmuch as they occur in English) tend to be pronounced "oo", "ah", and "euh", respectively, hasn't helped matters.
Additionally, prior to World War I, the history of English-French relations (and to a slightly lesser degree, but lasting up to World War II, French-German relations) can be summed up as a thousand years of trying to conquer each other, either through royal marriage or warfare, and this coloured a lot of English language fiction long before Hollywood (note that there's a fair number of British examples on this page as well). And the U.S.A. has a curious relationship with France as well — after all, both countries have fought democratic revolutions in roughly the same time, but went different roads afterwards.
It certainly doesn't help that due to Paris's worldwide prominence and the fact that French cities are so incredibly small in comparison (French cities haven't been evenly developed historically, so much so that Paris is the only city with more than two million people not counting its extended perimeter), France is often seen as a Land of One City whose inhabitants happen to be hurried and businesslike, in a similar way to New York, for the simple reason of being in one of the capital cities of the world. Actually, Parisians themselves are often disliked by the other French who think of them as the quintessential haughty French Jerks with a superiority complex (the word "province", dating back to the pre-Revolution Ancient Régime, is often pejoratively used by Parisians to summarize France as "Paris and the province"), just as certain Americans hate New Yorkers or other big city dwellers.
In fact, in France, this stereotype exists but is more of a Parisian jerk trope. Due to Paris being such an international juggernaut and being the "facade" of France this resulted in the unfortunate trope of France being a country of jerks.
There's also a very negative view of French tourists in any service industry, who are viewed as overly stingy (especially to service personnel who expect or live on tips- in fact, in France and many other European countries, it isn't customary to tip, because service is almost always included in the price) and overly cold and uptight.
