Short Answer: Just like a boxing match, damage adds up over time. Each turn your Rooster can take damage, and as long as the total damage they’ve taken doesn’t exceed their health, keep fighting.
Long Answer: Your Rooster is going to take damage from time to time. Attacks range in value from 1 to 4. Any Attack that isn’t dodged / cancelled, will deal damage to your Rooster. Keep track of that damage by simply keeping the Attack Cards with your Rooster. When the total damage your Rooster has taken meets or exceeds their Health (feathers), they are defeated, and go to the Egg Basket of the player who is responsible for their demise.