How did the Islamic conquest of Iran happen?

Before the Islamic conquest, Iran was under the rule of the
Sasanids, and suffered from a great deal of injustice, corruption and
oppression. Zoroastrianism was the official religion of Iran until the
Islamic conquest. 

The Islamic conquest of Iran began during the caliphate of
Abu Bakr as-Siddeeq (may Allah be pleased with him), when al-Muthanna ibn
Haarithah conquered some of the neighbouring lands in the region of
Mesopotamia in 13 AH. 

During the caliphate of ‘Umar (may Allah be pleased with
him), Abu ‘Ubayd ath-Thaqafi (may Allah be pleased with him) tried to invade
Iran from the south-west, and faced the Persians in the Battle of the Bridge
in 13 AH. He met Jaabaan in battle, and Allah defeated the Magians and
enabled the Muslims to capture Jaabaan. Then Abu ‘Ubayd marched to Kaskar
when he and Nursi met in battle and he defeated him. Then he met Jalinus in
battle and defeated him. Then the tide of fortune turned against the
Muslims, and many of them were killed and drowned, and Abu ‘Ubayd (may Allah
be pleased with him) was killed. 

Siyar A‘laam an-Nubala’
(2/421) 

Then the Muslims completed the conquest of Iran under the
leadership of Sa‘d ibn Abi Waqqaas in 14 AH, when the Muslim army in Persia
inflicted a mighty defeat (on the Persians) in the Battle of al-Qaadisiyyah,
which was one of the major battles of the Muslims. 

The historians stated that the Muslims were between seven and
eight thousand strong, and that Rostom, the Persian commander, had sixty
thousand troops. 

On a Monday in Muharram 14 AH, a strong wind blew and lifted
the tents of the Persians from their places, and the couch of Rostom, that
had been set up for him, was overturned. He hastened to ride his mule and
flee, but the Muslims overtook him and killed him. They also killed al-Jalinus,
the commander of the Persian vanguard, and the Persians were defeated –
praise be to Allah – to the last man and the Muslims pursued them. On that
day those who were chained were killed in their entirety, of whom there were
thirty thousand. Ten thousand were killed in the battle, and before that a
similar number had also been killed. On that day and on the previous days,
two thousand five hundred Muslims were killed, may Allah have mercy on
them. 

The Muslims pursued those who fled until they entered the
capital city behind them, namely al-Madaa’in (Ctesiphon), which was the seat
of the court of Chosroes. 

From the battle of al-Qaadisiyyah the Muslims seized an
indescribably huge amount of wealth and weapons. The booty was gathered
after giving the troops their shares, and the khums and glad tidings of the
victory were sent to Ameer al-Mu’mineen ‘Umar ibn al-Khattaab (may Allah be
pleased with him). 

Al-Bidaayah wa’n-Nihaayah
(9/630) 

Then the Muslims continued their advance into Persian
territory, and were enabled to conquer the southern part of Iran, whilst the
ruler Yazdegerd III retreated to the region of Isfahan in central Iran,
where he began to assemble troops in an attempt to retake what he had lost.
The Muslims and the troops of Yazdegerd met in the Battle of Jalula in 18
AH, which ended with the defeat of Yazdegerd and his army, and his retreat
towards Isfahan. 

At-Tabari (may Allah have mercy on him) said: 

On that day Allah killed one hundred thousand of them, and
the slain covered [jallalat] the ground in all directions. It was called
Jalula because it was covered with the bodies of their slain. 

Tareekh at-Tabari
(4/26) 

Then Yazdegerd met the Muslims in the decisive Battle of
Nahawand in 21 AH, where the Muslims won a clear victory, after which the
Sasanids no longer had any power. The Muslims seized a great deal of booty,
which led to them dubbing this battle “the conquest of conquests”. 

Ibn Katheer (may Allah have mercy on him) said: It was a
mighty battle of great significance and wondrous events, and the Muslims
used to call it the conquest of conquests. End quote. 

Al-Bidaayah wa’n-Bihaayah
(10/111) 

It took the Muslims ten years to gain control over all the
Iranian lands, because of their huge size and difficult terrain. 

One of the things that played a role in the spread of Islam
in Iran was the migration of many Arab tribes to Iranians land in which they
settled and mixed with the Iranians. 

For nearly nine centuries, Iran followed the madhhab of Ahl
as-Sunnah wa’l-Jamaa‘ah, until the Shi‘i Safavid state seized control of it
in 906 AH. 

See: 

Tareekh at-Tabari
by Ibn Jareer at-Tabari

Siyar A‘laam an-Nubala’
by al-Haafiz Shams ad-Deen adh-Dhahabi

Al-Bidaayah wa’n-Nihaayah
by al-Haafiz Ibn Katheer ad-Dimashqi

Al-Mufassal fi Tareekh al-‘Arab qabl al-Islam
by Dr Jawaad ‘Ali

Iran fi Zill al-Islam fi’l-‘Usooi as-Sunniyyah
wa’sh-Shi‘iyyah by Dr
‘Abd al-Mun‘im Hasanayn

Qaadat Fath Bilaad Faaris
– Mahmoud Sheet Khattaab

Iran wa Afghanistan
– Mahmoud Shaakir

Iran mundhu Fajr at-Tareekh hatta al-Fath al-Islami
by Dr Muhammad ‘Abd al-Qaadir Muhammad

And Allah knows best.