Fasting in Muharram was obligatory and these were the first fasts observed by the Muslims as a part of their religious activities .The Prophet Mohammed (P.B.U.H) has narrated several times about the importance of fasting in Muharram.
Each day of muharram fast is equal to 1 month of fasting. Fasting on the day of Ashura has been a tradition even before fasting was made obligatory upon the Muslims. When Hazrat Moosa (AS) and the people of Israel were saved from the cruelty of Pharaoh. Muharram is one of the four sacred months. Huma Jabri.

At-Tabarani reported that Ibn 'Abbas narrated that the Prophet said: "Whoever fasts one day during the month of Muharram will get a reward of fasting 30 days."

Reason for fasting on 9 th and 10 th of Muharram. And how many days do we fast in Muharram? I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger. It is one of the four sacred months of the year during which warfare is forbidden. It is noteworthy that Muslim scholars have stated that fasting of `Ashura’ is of three levels as follows: 1. fasting for three days, i.e., on the 9th, 10th, and 11th of Muharram, 2. fasting the 9th and 10th of Muharram… When the Companions were ordered by Prophet Muhammad (PBUH) to fast on Ashura, they showed their concern that the Jews and Christians also fast on that day. Fasting, Rules, Rites and Recommendations for the tenth day of Muharram, the Asura. Fasting on the day of Ashura is a wonderful deed and here you can read more about fasting in Muharram and the virtues of fasting on and around Ashura. The tenth of Muharram was a day which not only the pagans of Quraish, but also the Jews and the Christians of the time held in honor before the advent of Islam; and they commemorated this day by fasting.

The year consists of twelve months of which four are sacred: three consecutive months, Dhu’l-Qa’dah, Dhu’l-Hijjah and Muharram, and Rajab of Mudar which comes between Jumada and Sha’baan.” It was proven from the Prophet (peace and blessings of Allaah be upon him) that the best fasting after Ramadaan is fasting in the month of Muharram. According to Hadees e Pak.

Also, fasting the 9th of Muharram is highly recommended by the Sunnah.

There is a great reward for fasting on 9th and 10th of Muharram the day of Ashura. This is a fabricated narration as Shaykh Al-Albani classified it. time.
Fasting on the 10th of Muharram (‘Ashura) is the best fast after Ramadan. Muharram is a pious observance, and fasting, though not mandatory, is recommended. “Observe fasting on the day of ‘Ashura’, but differ from the Jews and fast one day before it and one day after it.” (Ahmad) “We are to fast on two days: the 9 th and 10 th of Muharram to distinguish ourselves from the Jews.” Answer: All perfect praise be to Allah, The Lord of the Worlds. The Tenth day of Muharram is known as the Day of Ashura.