What is the significance of fasting on 9th and 10th of Muharram?

When the Prophet (peace and blessings be upon him) settled in Madinah, there was a large Jewish community there. He noticed that the Jews fasted on 10th of Muharram. When he asked them about the purpose of their fast, they said that they were celebrating the day on which God the Almighty saved Prophet Moses from a great danger. The Prophet (peace and blessings be upon him) said that he [and the Muslims] were closer to Moses than the Jews and so he fasted that day. He continued to fast on the 10th of Muharram as a voluntary act of worship until his death. That year he said: "If I live till next year, I will fast on 9th of Muharram" i.e. he would fast on 9th and 10th of that month. It is most likely that he wanted to distinguish his fast in Muharram from that of the Jews, although the reason for fasting is the same.
And God the Almighty knows best.