How did Islam spread before the invention of the printing press?

Sufism is responsible for the proselytism and spread of Islam, first in the Middle East, then as far as Africa and South Asia. Sufism was a mystical form of Islam. The Sufi doctrine was established by certain mystics, and propagated widely by missionaries. It focused on one's personal experience with god and a divine love. Its message had popular appeal. Sufis used poetry rich with metaphors and allegories.

Until the mid 9th century, two hundred years after the establishment of Islam, Islam was a minority religion in the Middle East. Beginning around the late 9th century, and over the course of the 10th century, Sufis made the Middle East 90% Muslim. In the 11th century, the concept of Sufi fraternal orders, or missionaries, developed. This allowed Sufism to spread its influence. In the 1200's, missionaries spread as far as Africa and India. They were established in important locations, and presided over a certain region.

Islam was introduced to these neighboring regions by commerce. Sufis would usually come next and popularize the religion.

Sufism

Sufism- Britannica.com

Spread of Islam This is only a reference for conversion statistics in the Middle East

I focus on: How did Islam end up with a single Qur'an, without variant readings?

In the early years of Islam two key factors ensured the production of a single version, without variants, of a book, the Qur’an, for the whole of Islam.

The first factor was the existence of a single, centralised, controlling authority of the whole of the Muslim world with the power of the sword to enforce its will.

The second factor was the determined desire of that central authority to produce a single book (without variants) to try to maintain the unity of the Muslim world (presumably preferably under its own authority).

I write as a Bible believing Christian with an interest in all things religious. I have read all the Qur’an (“The Koran”, translated by N.J.Dawood, penguin classics), and a bit of the hadith literature. I shall give references at the end since they are not much more reading than this answer.

When Mohammed died in 632 CE the Qur’an as a book didn’t exist. The Qur’an is the sayings of Mohammed and the revelations purportedly given him by Allah. What did exist were these sayings and revelations written in various places: it existed on shards, on paper, on animal skins, and essentially on anything which could be written on. Because of widespread illiteracy in these early days, from the beginning, “the sayings” were memorised and recited by Muslims: memorisation and recitation have remained an important feature of Islamic devotion ever since. At the time of the production of the Qur’an in a single book the sayings were gathered together from all the possible sources. While some of these sayings were written down, others existed only in the minds of those who had memorised them; sometimes just a single person. Mohammed left no instructions for the production of a single book; some evidence suggests he may have been against the idea of producing it (Sahih Bukhari 6.61.509).

When Mohammed died Abu Bakr was chosen to succeed him as the first caliph. Some of the tribes that had been brought under Islam rebelled saying their allegiance had only been towards Mohammed. “Civil war” known as the Ridda Wars or the “Wars of Apostasy” broke out in the Arabian peninsula in 633 CE. Muslims were being killed in large numbers in battles to restore dominion. There was a fear amongst the leaders of Islam that large numbers of the sayings of Mohammed were going to the grave with the Muslims because a Muslim might be the only one who could remember a saying (Sahih Bukhari 6.61.510). Because of this, the one who became the second caliph, Uthman, ordered four men, who were most respected in matters relating to the sayings, to gather together all the sayings to produce a single book (Sahih Bukhari 6.61.507).

WHEN THE SINGLE BOOK, the first Qur'an, HAD BEEN PRODUCED, Uthman SENT A COPY OF THE NEW AUTHORISED VERSION OF THE QUR'AN TO EVERY MUSLIM PROVINCE and ordered that all the sayings of Mohammed on shards, paper, animal skin, etc, as well as all fuller collections of Mohammed’s sayings, BE BURNED. This is declared in the most reliable of the hadith, Sahih Bukhari (vol 6.book 61.510).

The hadith called Sahih Bukhari, in nine volumes, is considered the most reliable (but even individual statements in this may be legitimately doubted by a Muslim: after all, even the Sahih Bukhari is not inspired by God. The Muslim believes only the Qur’an is inspired.) Even so, there are six hadith which are considered reliable. After the Sahih Bukhari the ones considered the most reliable are the Sahih Muslim, then Sunan Abu Daud, then Sunan Tirmidhi, Sunan Ibn Majah, and Sunan Nisai in that order. These six are collectively called the Sahih Sittah. There are many more books in the hadith by other authors.

An English version of Sahih Bukhari is online and you can read it here:- https://www.sahih-bukhari.com/Pages/Bukhari_6_61.php

Volume 6, books 60 and 61 are about the Qur’an including how it was produced. Especially important is book 61.

"The Caliph 'Uthman ordered Zaid bin Thabit, Said bin Al-As, 'Abdullah bin Az-Zubair and 'Abdur-Rahman bin Al-Harith bin Hisham to write the Quran in the form of a book (Mushafs)" - Sahih Bukhari 6.61.507

6.61.509 needs to be read in full.

"Uthman sent to every Muslim province one copy of what they had copied, and ordered that all the other Qur'anic materials, whether written in fragmentary manuscripts or whole copies, be burnt." - Sahih Bukhari 6.61.510

I would recommend reading the penguin edition of the Qur’an, it is shorter than the New Testament. Then, you will know much more about a powerful influence on the present world. You will be in a much better position to decide if Islam is a religion of peace or otherwise. The Qur'an will either commend itself to you or maybe not.

Islam at first mostly spread by the sword. It could rapidly spread because of convenient circumstances. The everlasting border wars between the Byzantine and Sassanid empires hurt both empires seriously. They were not able to resist first Muslim border raids and later outright conquest.

If those two empires hadn't bled themselves white against each other, islam probably would not have become a major religion. It would have remained a local religion within the Arabian peninsular.

Another opportunity was religious intolerance of the orthodox (Byzantine) church and high taxation by the government. The Byzantine government was not very popular amongst some of its inhabitants. That is one of the reasons why Egypt quickly succumbed. Egyptians found anything better than the Byzantines. It took about one year to conquer all of Egypt.

Same story for the conquest of Spain. The ruling Visigothic kings were Arians. Their subjects were Spanish (or Iberian) Roman Catholics. The two didn't like each other very much. Withing the royal family a feud ran for kingship. One faction actually supported Tariq to lop off the other faction. Tarig was more than happy to oblige.

General Tariq was explicitly forbidding to invade, but he smelled an opportunity and took it. He took the Visigothic crown within one year. The population basically stood aside, with little support for their rulers.

Other religions used force as well. But within Islam it is doctrinal. Read the Koran. The sword is not a mere symbol, as the cross is.

As to how it spread so quickly without a written narrative:

Society back then was not much literate. Especially the Arabian peninsular was almost completely illiterate. Mohamed himself couldn't read or write. So oral traditions were very important.

During the first 2 centuries there was no written Koran as a directive body. Paper was relatively rare and expensive. Some early Koranic texts were written on palm leaves, camel bones, stones and shards. Around 650 AD the first attempts were made to codify the Koran. That took about 200 years to be finalized.

It's not remarkable at all that a religion quickly spreads by word of mouth. A lot of early Christians were also illiterate. A lot, if not the majority of religions in history, spread by word of mouth.

Islam never relied on printed materials to spread itself. The main method of spreading Islam was not printed scriptures, but manned preaching.

There were (and still are) a group of people among Muslims, called Hafiz, who would memorize Quran. Their only objective was to retain Quran through memorization, and convey it to others.

More reading: Spread of Islam