Allah’s sacred month of Muharram is a blessed and important month. It is the first month of the Hijri calendar and is one of the four sacred months concerning which Allah says:

إِنَّ عِدَّةَ ٱلشُّهُورِ عِندَ ٱللَّـهِ ٱثْنَا عَشَرَ شَهْرًا فِى كِتَـٰبِ ٱللَّـهِ يَوْمَ خَلَقَ ٱلسَّمَـٰوَٰتِ وَٱلْأَرْضَ مِنْهَآ أَرْبَعَةٌ حُرُمٌ ۚ ذَٰلِكَ ٱلدِّينُ ٱلْقَيِّمُ ۚ فَلَا تَظْلِمُوا۟ فِيهِنَّ أَنفُسَكُمْ
Verily, the number of months with Allah is twelve months (in a year), so was it ordained by Allah on the Day when He created the heavens and the earth; of them four are Sacred. That is the right religion, so wrong not yourselves therein. [At-Tawbah 9:36]

Muharram is so called because it is a sacred (Muharram) month and to confirm its sanctity. Allah’s words “so wrong not yourselves therein.” mean do not wrong yourselves in these sacred months, because sin in these months is worse than in other months. It was reported that Ibn ‘Abbas (RA) said that this phrase (so wrong not yourselves therein.) referred to all the months, then these four were singled out and made sacred, so that sin in these months is more serious and good deeds bring a greater reward. Qatadah (rahimahullah) said concerning this phrase (so wrong not yourselves therein.) that wrongdoing during the sacred months is more serious and more sinful than wrongdoing at any other time.

The Hadeeth which encourages us to observe more nafl fasts during Muharram also gives us a further understanding of the virtue of this month. Abu Hurairah (RA) narrates: “the Messenger of Allah ﷺ said:
«أفضل الصيام ، بعد رمضان ، شهر الله المحرم» رواه مسلم
‘The best of fasting after Ramadan is fasting Allah’s month of Muharram.’ [Muslim, 1982]

The phrase “Allah’s month”, connecting the name of the month to the name of Allah in a genitive grammatical structure, signifies the importance of the month. Al-Qaari (rahimahullah) said: “the apparent meaning is all of the month of Muharram.”, but it was proven that the prophet (peace and blessings of Allah be upon him) never fasted any whole month apart from Ramadan, so this hadeeth is probably meant to encourage increasing one’s fasting during Muharram, without meaning that one should fast for the entire month.

Why do we fast?
عَنْ أَبِي مُوسَى ـ رضى الله عنه ـ قَالَ دَخَلَ النَّبِيُّ صلى الله عليه وسلم الْمَدِينَةَ وَإِذَا أُنَاسٌ مِنَ الْيَهُودِ يُعَظِّمُونَ عَاشُورَاءَ وَيَصُومُونَهُ فَقَالَ النَّبِيُّ صلى الله عليه وسلم ‏ “‏ نَحْنُ أَحَقُّ بِصَوْمِهِ ‏”‏‏.‏ فَأَمَرَ بِصَوْمِهِ‏.
Narrated Abu Musa (RA): When the Prophet ﷺ arrived at Medina, he noticed that some people among the Jews used to respect Ashura’ (i.e. 10th of Muharram) and fast on it. The Prophet ﷺ then said, “We have more right to observe fast on this day.” and ordered that fasting should be observed on it. [Bukhari]

In fact the fasting of Muharram was obligatory before the fasting of Ramadaan.
عَنِ ابْنِ عُمَرَ ـ رضى الله عنهما ـ قَالَ صَامَ النَّبِيُّ صلى الله عليه وسلم عَاشُورَاءَ، وَأَمَرَ بِصِيَامِهِ‏.‏ فَلَمَّا فُرِضَ رَمَضَانُ تُرِكَ‏.‏
‏Narrated Ibn `Umar (RA): The Prophet (ﷺ) observed the fast on the 10th of Muharram (‘Ashura), and ordered (Muslims) to fast on that day, but when the fasting of the month of Ramadan was prescribed, the fasting of the ‘Ashura’ was abandoned.