Partial answer: I can't tell you when, but can tell you why, because I speak or understand several European languages. In most of them, you don't say "September 11." You say (The) Eleventh of September." In quite a few languages, the "the" is omitted, and in several the word "of" just becomes an ending on the word month. In Russian, for example, it's enough to say odynadtsyatiy sentyabra. No "the", no "of." Now, why do they do it that way? The internal logic of English is different from the internal logic of a language where most of the words change their spelling as they change their function. In English, the meaning of a word is determined by its position in a sentence, because English is a distributive language. You can't really say "Throw me out the door my hat" in English, but in most European languages you can--because the words "me", "door", "my" and "hat" would change their spellings to show that you meant "Throw my hat to me through the door." In most European langauges, to say "September 11" would not convey any information. You would have to change the ending on both words, and perhaps add the preposition "of", to show that the two words were in a relationship, and that what you meant was not the number 11, but the eleventh day, and that it was the eleventh day OF September. Of course, the day will probably come when everyone switches to the logical way of doing it--2001/09/11, which automatically sorts dates on your computer. That's the way I date all my mail, and nobody has yet complained.

Roger's partial answer makes sense, but the other two answers make sense too. This question has puzzled me for a while now. I'm from the US, and I live in Europe for the past 10 years now, and I am used to the DD/MM/YY format. My puzzlement derives from the fact that on copies of the US Declaration of Indepence, the date is in MM/DD/YY (or rather, written out "July 4th 1776"). So does this means the US has been using such a format since then? If our American fore-fathers were of European descent, and assuming Europeans back then used DD Month Year format, why would the Declaration of Indepence use Month DD Year format? Or, was the world using Month DD Year format, and then at some point the rest of the world changed, but not the US? Similar to how most of the world is using the metric system while the US isn't? Or is it just a matter of English vs. non-English as Roger suggests?

Roger's answer doesn't take into account that England used the same dating format as non-English speaking countries. It would be natural for the early American colonists to continue using the familiar DD/MM/YY format and they did. So when and why did the Americans change their date format?

It began with the declaration of independence and was a symbolic rebellion against the British.