Davos survived the green fire and Tyrion's chain after falling off his ship by swimming under the water. He ended up washed up on a rocky outcrop. He was ready to give up on life, since his sons are all dead as far as he knows, and he spends most of the chapter in various states of nearly dying. Then he decided that Melisandre and her weird magic and her god are probably to blame for all this bad luck. Now he has a purpose, presumably to kill her. A passing ship comes to the rescue, and luckily for him it is crewed by men loyal to Stannis.