The traditional chef's uniform the coat and hat - the toque - is generally credited to Marie Antoine Careme who served as the chef de cuisine to noted French diplomat and gourmand Charles Maurice de Talleyrand-Perigord. Carme is pretty much the founder of the modern notion of the chef and is every bit as foundational to modern French cuisine as Goethe is to the German language.

As as aside, Talleyrand was a fairly major player is the French Revolution: he was a coauthor of the Declaration of the Rights of Man and was later a player in events that brought about the restoration of the Bourbon monarchy after the fall of Napoleon.

Prior to Careme's innovation chefs dressed in a manner not dissimilar to the priesthood. What began in Careme's kitchens was later spread to London and the Americas by Auguste Escoffier.

As other posters have observed, Careme's rational for the coloration was that it denotes cleanliness. The fact that it's easily bleached is just a happy coincidence.

This article from the National Culinary Review is a fairly good source on this until someone can drum up a primary source: http://www.cheftalk.com/a/jackets-and-toques-the-history-of-the-chef-uniform

So there are two different parts to this answer because the typical uniform for chefs includes the chef's hat, the "toque blanche" or just "toque", and the chef's jacket.

The toque in the form of a white, brimless, tall hat with folds at the top originated in the 19th century in France, and was adapted from similar hats worn by chefs for centuries. This type of hat was combined with a white double-breasted coat to form the traditional chef's uniform. So the hat has a longer history than the coat, but they were adapted and combined in 19th century France by Marie-Antoine Careme, a popular french chef. The design spread throughout Europe and remains popular to this day in many high-end restaurants. The hat itself can indicate who is the head chef in a kitchen.

Now why white?

Because it signifies cleanliness. Keep in mind that you'd wear an apron over the jacket to prevent most stains. If that failed you could reverse the jacket to cover stains in the middle.

The chef's uniform typically includes darker colored or patterned pants which would conceal stains better.

Edit 1: I could have included the history of the apron, similar hats worn by chefs, the scarf, and types of pants and shoes, but I decided to keep it simple and just talk about the toque and jacket.

Edit 2: If you can read french "L'art de la cuisine Française au dix-neuvième siècle: Tome 3" credits him with combining the toque and jacket and discusses some of his other contributions, however a broader history of chefs and the uniform can be found in "Cuisine and Culture: A History of Food and People" which contains much of the same information and more.

Definitely a good read for anyone who wants to get better at cooking. Gaining an appreciation for the rich history of cooking and the variety of specialized techniques used by chefs who are nothing short of genius will help you improve your own skills and appreciate every meal just a little bit more.

Cooking is one of the oldest human traditions, and is one of the things that really makes us human.

A note to would-be answerers:

This is /r/AskHistorians. Answers are expected to be thorough, comprehensive, and historical. A full six comments have been removed for being one-line notes about how white clothing shows stains and can be bleached. Further comments of that sort will result in temporary bans.

The source is Carême, Le Maître d'hôtel français, (Paris: Firmin Didot, 1822). The first volume is digitized here. He contrasts the old and the new chef in the [frontispiece] (http://www.pazzobooks.com/pictures/CAT000191_3.jpg) to this book. He discusses the whiteness of the bonnet as a sign of cleanliness, and the reasoning for the design of the chef's bonnet vol II, pp. 279-280.

See also Priscilla Parkhurst Ferguson's brief discussion in "Writing Out of the Kitchen: Carême and the Invention of French Cuisine," Gastronomica: The Journal of Food and Culture, Vol. 3, No. 3 (Summer 2003), pp. 40-51.