Wellington was renown for being on the front line with his men to rally them. He rode his horse all up and down the line while being shot at. Eton was his grammar school where, supposedly, he played games and became such a vigorous athlete warrior.

But, he never said it.

Eton didn't even have playfields or organized games. Even if it did, Wellington probably wouldn't have played them. He was very shy and reserved, and also quite thin and small (not an athlete by any means). Wellington didn't like Eton. He did, however, love the garden at the house he grew up in. 34 years after he left Eton he returned to the house to inspect the gardens. He said to his companion, "I really believe I owe my spirit of enterprise to the tricks I used to play in the garden."

Your quote first appeared 3 years after he died and were written by French writer and parliamentarian Count de Montalembert who visited Eton and the house for material for a book on the political future of England. He wrote, "One understand the Duke of Wellington when, revisiting during his declining years the beauteous scenes where he had been educated, remembering the games of his youth, and finding the same precoucious vigour in the descendents of his comrades, he said aloud, 'C'est ici qu'a été gagnée la bataille de Waterloo.'" (translated: Here is where the Battle of Waterloo was won) There was no mention of "playfields." It was later that the quote "The Battle of Waterloo was won on the playing fields of Eton" was attributed to him.