ANSWER:
The Jews of Madinah used to fast on the 10th of Muharram because Allah saved Moosa alayhissalam from the Pharoah and his people on this day. Prophet salla Allahu Alaihi wa Sallam said we have more right on Moosa therefore he started fasting this day and said if he were to be alive the following year, he would fast 9th of Muharram as well to be different from the Jews.