I mean restaurants that are known by their association with an ethnicity, eg. Chinese, Thai, Italian, etc.? When did people start saying things like "How about Chinese tonight?"
A good portion of the type and distribution of ethnic restaurants in the US has to do with immigration. Many of the common ethnic-themed restaurants correspond to large immigrant groups (Mexican, Italian, Chinese) and these restaurants often homogenize a very diverse food culture through their menu. Furthermore, these menus often substitute local food in place of harder to find ingredients from the old country or even invent "traditional" food that are actually quite new. In the latter case, the most famous example is the fortune cookie whose modern form originated in the West coast from Japanese and later Chinese restaurants.
The homogenization process frequently overlaps with the immigrant community's origins. For example, a lot of a "typical" US Chinese menu derives from Cantonese or Szechuan cooking, the regions where a the majority of Chinese immigrants came from prior to the 1990s. Moreover, being a restaurateur was a viable and lucrative business model to many of these immigrant groups. For Chinese immigrants, owning a restaurant was one of the few ways around the various exclusion acts that prohibited Chinese movement within the US.
Metropolitan cities, especially with their heterogeneous culture and milieu, are vital for many ethnic-themed restaurants to gain ground among consumers outside the ethnic community. In the words of the sociologist Alan Warde, the metropolitan public sphere (things like magazines, newspaper reviews, daily experience) is vital for "routinization of the exotic." The advent of mass culture allowed for the metropolitan experience to be understood far outside the confines of cities.
Sources
Kalb, Don. The Ends of Globalization: Bringing Society Back In. Lanham, Md: Rowman & Littlefield, 2000.
Shortridge, Barbara G., et al. The Taste of American Place A Reader on Regional and Ethnic Foods. Lanham: Rowman & Littlefield Publishers, 1999.
Warde, Alan. Consumption, Food, and Taste Culinary Antinomies and Commodity Culture. London: Sage Publications, 1997.