According to Mexican tradition there was a man named Juan Mendez who used a donkey to transport his self and the food to his taco street stand. This occurred during the Mexican Revolution (1910-1921) in the Belle Vista neighborhood of Ciudad Jurez, Chihuahua. He wanted to be able to keep the food warm that he was to sell, so he decided to wrapped the food in large homemade flour tortillas. He then wrapped them in napkins.
Even back then, they were a hit with the local townsfolk, so much that people would cross the border just to buy the "burritos". The word spread though, causing people from all over Northern Mexico to sell burritos.
Another thing to realize is that the authentic mexican burritos aren't like the burritos you find at most of the American restaurants. Authentic Mexican burritos generally only have two ingredients in each burrito and are wrapped in a thin flour tortilla. Some of the ingredients to choose from would have been: some form of meat, beans, chile rajas, potatoes, and asadero cheese.