Short answer
Because the Hijrah, the Prophet's ﷺ migration from Makkah to Madinah in 622 CE, turned the Muslims from a persecuted few into a community and state. When Umar set up the calendar around 16 to 17 AH, the Companions chose the Hijrah as its starting point, and Muharram as its first month.
The Sunni historical account, from Sahl ibn Sa'd in Sahih al-Bukhari, is that the Companions counted years from the Prophet's ﷺ arrival in Madinah, not from his birth, mission or death. This was settled in the caliphate of Umar.
The four schools agree on this.
Explanation
In Umar's caliphate, letters and records needed dates. Reports say Abu Musa al-Ash'ari wrote that letters came from Umar without a year on them. Umar gathered the Companions. Some suggested dating from the Prophet's ﷺ birth, others from his mission or his death. They chose the Hijrah, because it was the moment that separated truth from falsehood and the start of the Muslim community. Sahl ibn Sa'd said they did not count from his mission or his death, only from his arrival in Madinah.
They then chose Muharram as the first month, because it came after the Hajj and the pledges of Aqaba that prepared the way for the Hijrah, even though the Prophet ﷺ actually arrived in Madinah in Rabi al-Awwal. So year 1 AH began in July 622 CE.
Why is the Hijrah so important? It was the end of thirteen years of persecution in Makkah and the start of the first Muslim society in Madinah: the first mosque, the brotherhood between the Muhajirun and Ansar, the adhan, the fasting of Ramadan and the zakat. Allah praises those who emigrated and those who gave them shelter.
For the story of the Hijrah itself, see What Is Hijrah in Islam? For the calendar today, see Islamic Calendar and Islamic New Year.
The evidence
Quran, Surah At-Tawba 9:40
إِلَّا تَنصُرُوهُ فَقَدْ نَصَرَهُ ٱللَّهُ إِذْ أَخْرَجَهُ ٱلَّذِينَ كَفَرُوا۟ ثَانِىَ ٱثْنَيْنِ إِذْ هُمَا فِى ٱلْغَارِ إِذْ يَقُولُ لِصَٰحِبِهِۦ لَا تَحْزَنْ إِنَّ ٱللَّهَ مَعَنَا ۖ فَأَنزَلَ ٱللَّهُ سَكِينَتَهُۥ عَلَيْهِ وَأَيَّدَهُۥ بِجُنُودٍۢ لَّمْ تَرَوْهَا وَجَعَلَ كَلِمَةَ ٱلَّذِينَ كَفَرُوا۟ ٱلسُّفْلَىٰ ۗ وَكَلِمَةُ ٱللَّهِ هِىَ ٱلْعُلْيَا ۗ وَٱللَّهُ عَزِيزٌ حَكِيمٌ ﴿٤٠﴾
If you do not aid the Prophet - Allah has already aided him when those who disbelieved had driven him out [of Makkah] as one of two, when they were in the cave and he said to his companion, "Do not grieve; indeed Allah is with us." And Allah sent down his tranquillity upon him and supported him with angels you did not see and made the word of those who disbelieved the lowest, while the word of Allah - that is the highest. And Allah is Exalted in Might and Wise.
The Hijrah and Allah's help in the cave.
Quran, Surah Al-Anfaal 8:72
إِنَّ ٱلَّذِينَ ءَامَنُوا۟ وَهَاجَرُوا۟ وَجَٰهَدُوا۟ بِأَمْوَٰلِهِمْ وَأَنفُسِهِمْ فِى سَبِيلِ ٱللَّهِ وَٱلَّذِينَ ءَاوَوا۟ وَّنَصَرُوٓا۟ أُو۟لَٰٓئِكَ بَعْضُهُمْ أَوْلِيَآءُ بَعْضٍۢ ۚ وَٱلَّذِينَ ءَامَنُوا۟ وَلَمْ يُهَاجِرُوا۟ مَا لَكُم مِّن وَلَٰيَتِهِم مِّن شَىْءٍ حَتَّىٰ يُهَاجِرُوا۟ ۚ وَإِنِ ٱسْتَنصَرُوكُمْ فِى ٱلدِّينِ فَعَلَيْكُمُ ٱلنَّصْرُ إِلَّا عَلَىٰ قَوْمٍۭ بَيْنَكُمْ وَبَيْنَهُم مِّيثَٰقٌۭ ۗ وَٱللَّهُ بِمَا تَعْمَلُونَ بَصِيرٌۭ ﴿٧٢﴾
Indeed, those who have believed and emigrated and fought with their wealth and lives in the cause of Allah and those who gave shelter and aided - they are allies of one another. But those who believed and did not emigrate - for you there is no guardianship of them until they emigrate. And if they seek help of you for the religion, then you must help, except against a people between yourselves and whom is a treaty. And Allah is Seeing of what you do.
Allah's praise of the emigrants and those who sheltered them.
Sahih al-Bukhari 3934
Sahih... ما عدوا من مبعث النبي صلى الله عليه وسلم ولا من وفاته ...
"They did not count from the Prophet's mission nor from his death."
The Companions dated the calendar from his arrival in Madinah.
Where the ruling is recorded
- Fath al-Bari (Ibn Hajar), Commentary on Kitab Manaqib al-Ansar, chapter on the dating, from when they dated. Gives Umar's consultation and the choice of Muharram.
- al-Bidaya wa al-Nihaya (Ibn Kathir), Events of the year 16 AH, the setting of the Hijri calendar. Ibn Kathir's account of the calendar.
- Sirat Ibn Hisham, Account of the Hijrah of the Messenger of Allah. The migration itself.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.