Of course, the change from Old English to Middle English did not occur all at once. So there is no one date on which Old English died and Middle English began. The best we can say is that Middle English came about because of the Norman Conquest of England. This happened in 1066.
After 1066, the French spoken by the Normans started to change the English language. This did not happen all at once. Even so, the French that the Normans spoke came to transform the English language. It brought in a great deal of new vocabulary. It also moved English away from being a language with all sorts of cases and inflections and moved it toward being the simpler language we now use.
When did Anglo-Saxon English change to Middle English?