None of the sources I’ve found can date this before 1975. The consensus is that when casinos in Las Vegas were in a particularly competitive period they would vie for customers just as banks offer free toasters to people who open new accounts. Back then a chicken dinner in the casino business was priced at just under $2.00. In those days the most common bet was $2.00, and in an attempt to attract players to place bets, the stickman at roulette or craps would shout the phrase. In short, it was shilling for players at an underpopulated table by telling wandering innocents that they could win a chicken dinner by playing.
* Just found somebody who remembers that it was used in much the same way at The Aqueduct Racecourse in New York in 1954. The most common bet on a horse was two bucks and even a pretty bad payoff would provide you the price of the restaurant’s chicken dinner, priced on the menu at about a dollar. In either case, everyone seems to remember that the chicken dinners were not only the least expensive meals on the menus, but were uniformly awful.
“Winner Winner, Chicken Dinner” has now become a catchphrase for anyone who wins in sports, gambling, trivia, and other pastimes, especially someone who wasn’t expected to win. FWIW, I am deeply suspicious about this origin of the phrase, but a chicken dinner is a powerful incentive.