The festival takes place on the beach between 10am and 6pm on the 7th of Summer. On the 6th, Popuri will come to your farmhouse in the morning to ask what chicken you'd like to enter for the contest. If you own a chicken and want to enter, choose a chicken and Popuri will take it for the festival on the following day. Popuri will not take chicks or sick chickens. If you do not own a chicken, Popuri will still encourage you to come watch the festival.
On the 7th, go to the beach and talk to Popuri to start the judging of the contest. Your chicken will be up against three other contestants. The higher the heart level your chicken has, the better your chance of winning. A chicken at 9-10 hearts will almost be guaranteed to win (although 9 hearts may still get you second place). Any placing besides first place counts as a loss. Nothing will happen if you lose.
If your chicken places first, it's level will increase.