This quote comes from the end of Chapter Four and refers to Jack's mask that he paints on his face to hunt the wild pig on the island. What is particularly interesting about it is that the quote seems to suggest that Jack becomes a different person with the paint on his face. His reaction to seeing his reflection in the coconut shell full of water is to jump about, "laughing excitedly." There is a sense in which Jack is thrilled to have esaped the narrow taboos of British society in terms of decorum and order.
The irony of this phrase lies in the fact that the reflection is no actual "awesome stranger" but is only Jack all along. Jack imagines it to be somebody or something external to himself. He very quickly realises that the savage barbarian he has turned into is not actually external to him at all and is a reflection of his own heart and soul. Golding thus uses this quote to ironically comment upon the way in which we think we are so cultured and sophisticated, whereas in reality we are all brute savages dressed up with nothing but a thin veneer of civilisation.