Why is the month Muharram so important?
Answer:
The leader of the months is Muharram
Our Beloved Prophet Muhammad (pbuh) states that:
“The leader of the people is Prophet Adam, of the Arabs is Prophet Muhammad (pbuh), of the Roum is Suhayb (ra), of the Persian is Salman (ra), the leader of the trees is Sidra, of the months is Muharram, of the days is Friday, of the words is Quran, the leader of the Quran is Surah al Baqara and its leader is Ayat-al Kursi.” (Imam Suyuti)

Muharram is the first of the lunar months
In the first years of Islam, the infidels and polytheists used to torture Prophet Muhammad (pbuh) and His distinguished companions to hinder Islam, Islamic life and the spread of Islam. Their tyranny arose to such an extent that the Muslims were exposed to starving through brutal embargos. So, upon Allah’s order, Muslims emigrated from Makka to Madina to continue their Islamic life and mission. This is called “Hijra”.

Years after the Hijra, in the Caliphate time of Hadrat Umar (ra) a commission composed of the sahabas (the distinguished companions of Our Prophet) accepted the day of Hijra, because of its being highly precious, as the first day of the calendar and the New Year. Thus, every Muslim considers the importance of Hijra, the holy journey displayed not for self desires but displayed only for the consent of Allah (swt) in every new year.

The Month of Muharram was an important month even before Islam
Muharram was the first month of the year even before Islam. It was highly esteemed, in the pre-Islamic age of ignorance. Even before Islam, Arabs regarded it as haram to wage war on the months Dhul-Qaʿda, Dhul-Hijjah and Muharram. To such an extent that, if they had to war in one of these months, somehow, they replaced the months in order not to disrespect the month.

Muharram is the anniversary of many significant events for Islam and for humanity
As narrated, Allah (swt) created the earth and the heaven on this holy month and bestowed many special bounties to His prophets on the Day of Ashura, the 10th and the most important day of Muharram. Since the Ashura, which is quite precious that many significant events happened and Allah accepted many prays on this day, takes place in the month of Muharram, it improves the importance of Muharram.
The first ten days of Muharram are more important and virtuous than the rest
Our Prophet Muhammad (pbuh) informed His ummah of the importance of the first ten days of Muharram and advised to utilize them well:

“O my companions and my ummah! Be careful on the first ten days of Muharram! Do not fail in respect for these days. Fasting and doing the other good deeds have a lot of thawabs on these days. Allah (swt) created the earth and the heaven in the first ten days of Muharram. Futhermore, the end of the world will come on those days. Allah (swt) saved Prophet Nuh from the flood and the infidels drowned on the first ten days of Muharram." (Gunyat-ut Talibin)