A popular story about the origin of this phrase is that in Las Vegas you could get a nice chicken dinner for less than $2.00. Since winning an average bet would make you over $2.00, the two were equated as being approximately the same value. If you won, you could buy a chicken dinner, and I'm sure that the rhyming factor didn't hurt for people who were rubbing in their win. :)
The phrase "Winner Winner Chicken Dinner" means that the winner from the given bet has to buy the chicken. It is derived from a legend that some years ago in Las Vegas, where there was a three-piece chicken dinner with a potato and a veggie went for $1.79. A bet would go for $2 and in case you won the bet, you had enough to buy the dinner.