The modern concept of a dedicated restaurant [in the European context] dates back to the period just before and most certainly immediately after the French Revolution. Prior to this, most places in which one could eat outside of the home were pubs or inns, which provided food as an auxiliary service, rather than the primary fare with dedicated professional staff.

Classical French [haute] cuisine started with the downfall of King Charles I of England, with whom the Protestants clashed and ultimately brought his downfall. The French, Catholic, loyalists understood the threat to their king and the Church and an unidentified group of them bound together to help fight back in a cultural way by writing the first classic French cookbook, Le Cuisinier Francoise, commonly ascribed to Francoise Pierre La Varenne.

This book and its surrounding appreciation of the new haute cuisine led to something of a cross-channel "arms race" of sorts during which the Nobility would compete to see who could hire the best chefs and throw the most lavish parties. The number of chefs and full cooking staff employed in England and France grew tremendously.

Then the French Revolution struck and overnight, most of the highly trained chefs were unemployed. They needed new work and found themselves in great demand as cooks in public houses, and started opening restaurants. The idea took hold in France first, and grew in popularity throughout Europe.

Source: Acquired Taste: The French Origins of Modern Cooking, by Sarah Peterson.

Also, over a decade of professional cooking and study of the trade.

edit: For those who might argue with my use of "Classical" to describe haute cuisine, that is correct as true Classical Cuisine is ascribed to Escoffier. However, the break from Ancien Regime is Le Cuisinier Francoise.