Why fast on Ashura?

"The Messenger of Allah (peace and blessings of Allah be upon him) said: 'The best of fasting after Ramadhan is fasting Allah's month of Muharram.'  (Muslim) 
This hadith is meant to encourage increasing one's fasting during Muharram, without meaning that one should fast for the entire month; which many tend to misunderstand.
Ashura takes place in the month of Muharram and commemorates the day that Allah(SWT) saved the Children of Israel (AS) from Pharaoh; the victory of Prophet Musa (AS)
And We took the Children of Israel across the sea, and Pharaoh and his soldiers pursued them in tyranny and enmity until, when drowning overtook him, he said, "I believe that there is no deity except that in whom the Children of Israel believe, and I am of the Muslims." (Surah Yunus:90) 
Fasting on the 10th of Muharram, widely known as the Day of 'Ashura', expiates for the sins of the past year. When Prophet Muhammad (SAW) arrived in Madinah in 622 CE, he found that the Jews also fasted on the 10th of Muharram. He asked them the reason for their fasting on this day, to which they said:
"This is a blessed day. On this day Allah saved the Children of Israel from their enemy (in Egypt) and so Prophet Musa [Moses] fasted on this day giving thanks to Allah." 
The Prophet (SAW) then said,
"We are closer to Musa than you are." (Al Bukhari) 
He fasted on that day and commanded Muslims to fast on this day. The following year, Allah commanded the Muslims to fast in the month of Ramadan, and so, fasting on the day of 'Ashura' became optional. It is also reported that the Prophet Muhammad (SAW) intended to fast on the ninth and tenth. Ibn 'Abbas reported: The Messenger of Allah fasted on the day of Ashoora and ordered the people to fast on it. The people said,
 
"O Messenger of Allah, it is a day that the Jews and Christians honor." The Prophet Muhammad (SAW) said, "When the following year comes, Allah willing, we shall fast on the ninth." (Muslim and Abu Dawud) 
However, the death of the Prophet Muhammad (SAW) came before the following year. Earlier, the fast on the 10th of Muharram was intended but then 9th of Muharram was as well. The reason why the fast of Ashura (10th of Muharram) should be kept along with one on 9th or 11th Muharram was to distinguish Muslims from the Jews.
Thus, according to scholars, one may choose to fast 'Ashura' on all the three days (9th, 10th, and 11th ) or, the two days (9th and 10th ) or (10th and 11th ). The Prophet Muhammad (SAW) said:
 
"Fasting the day of 'Ashura' (is of great merits), I hope that Allah will accept it as an expiation for (the sins committed in) the previous year." (Muslim) 
This expiation is in regards to the minor sins one commits comes, not the major ones; there still is a need for repentance. It is a misunderstanding to rely on fasting on Ashura to "wipe your slate clean as it alone will not atone for all the sins.
The scholar Ibn Al-Qayyim Al-Jawziyya initiated: This misguided person does not know that fasting in Ramadan and praying five times a day are much more important than fasting on the Day of Arafah and the Day of 'Ashura', and that they expiate for the sins between one Ramadan and the next, or between one Friday and the next, so long as one avoids major sins. But they cannot expiate for minor sins unless one also avoids major sins.
All in all, one should take advantage of this opportunity to bring oneself closer to Allah(SWT) on 'Ashura' by fasting and praying for forgiveness. May all the Muslims sustain this practice every year.