Letters would come to him, and he would not know when they were written and when they were sent. Therefore, he (رضي الله عنه) set the calendar to make it begin from the month of al-Muḥarram.
Shaykh Ṣāliḥ al-Fawzān (حفظه الله) said:
As for the beginning of the year [starting the Islamic calendar with al-Muḥarram], then it is an affair that was adopted in the era of ʿUmar [ibn al-Khaṭṭāb, the second rightly guided Khalīfah] (رضي الله عنه) because of the letters that would come to him from his workers.1
Letters would come to him, and he would not know when they were written and when they were sent. Therefore, he (رضي الله عنه) set the calendar to make it begin from the month of al-Muḥarram, since it was the month in which the Ḥajj pilgrims would set out to return to their countries.
[Furthermore,] the most beloved month to Allāh after Ramaḍān is Allāh’s month, al-Muḥarram; therefore, the calendar began from Muḥarram.
Another reason was that even though the Hijrah occurred during the year, ʿUmar wanted to start the calendar from the beginning of Muḥarram because it was a sacred month2 and because the Ḥajj pilgrims would set out upon the backs of camels during Muḥarram [to return to their countries].
 Due to the problems that arose, ʿUmar ibn al-Khaṭṭāb gathered the Companions to consult with them on the issue of the Islamic calendar and to decide which of the months should be its first. They decided to start the calendar from the month of Muḥarram, as the shaykh clarified.  Muḥarram is one of the four sacred months. The other three are: Dhū al-Qaʿdah, Dhū al-Ḥijjah and Rajab.
The Prophet (ﷺ) said: “The year is twelve months, of which four are sacred: the three consecutive months of Dhū al-Qaʿdah, Dhū al-Ḥijjah and Muḥarram, and Rajab Muḍar, which comes between Jumādá and Shaʿbān.” Reported by al-Bukhārī.