What is the reference to the playing fields of Eton in chapter 8?
This allusion comes when Mr. Ludsbury sees Gene and Finny exercising late at night; or, more accurately, Gene exercising and Finny coaching him. Gene remarks that ordinarily, they would be in trouble, but WWII had changed the atmosphere of Devon.
There was no rule explicitly forbidding exercise at such an hour, but it was not expected;ordinarily therefore Mr. Ludsbury would have disapproved. But the war had modified even his standards; all forms of physical exercise had become conventional for the Duration.
Thus, Finny and Gene get away with their training, even at a late hour, because all boys were expected to prepare for war. Any form of physical activity was seen as a viable program to train soldiers. But Mr. Ludsbury does not let them get away without moralizing. He makes it clear that the war is the only reason he's allowing them to continue:
"Games are alright in their place," he said, "and I won't bore you with the Eton Playing Fields observation, but all exercise today is aimed of course at the approaching Waterloo. Keep that in your sights at all times, won't you."
As the previous poster explained, these allusions set the story and WWII in the context of other British battles. It shows that the boys are being trained to accept war as something historic and patriotic, & to be willing to give up their lives for their country. Thus, even sports, games, and exercises must be aimed at contributing to the war effort.
The reference here is to a famous quote from Arthur Wellesley, the Duke of Wellington. He was the leader of the forces that defeated Napoleon at the Battle of Waterloo in 1815. This is one of the most famous battles in British history.
Wellington is supposed to have said (though people now say he probably didn't actually say it) that the battle was won on the playing fields of Eton. Eton is a famous private boarding school for upper class kids in England. The idea behind the quote is that the British learned how to be brave and tough by playing sports at their boarding schools.