During the holy month of Ramadan, Allah gave the Holy Quran. One day, Muhammad had a vision in which the angel Jibril told him how Allah wanted people to live.
The Allah s words were revealed to Muhammad through the angel over a period of 23 years, and the words were written down to form the Quran. Ramadan is celebrated to remember this very special month.
Also, if some other traditions are to be believed. Muhammad saw people celebrating two days with fun and merriment. He them proposed two other days granted by God to Muslims. This became a legend after that.
Another legend says, Muhammad d father in law asked some singing girls to stop when Muhammad said that it was our Eid. And allowed the girls to sing.