The Quran is very unique in the sense that in the beginning, it wasn't actually a book as we have it now but it was memorized orally.
Nowadays most people are literates and we like to see things in writing, but in past societies, people were generally not literate. For instance, in the time of the Prophet Muhammad, peace be upon him, most people were not literate. Even in our literate culture now, you have lectures being presented orally. Why don't professors write their lectures on paper and hand it out to all of the students?
Well, you can do that, but there is a certain place in the human mind for oral presentation. People want to hear it and the Quran was heard by people. God revealed the Quran in such a way that it would be recited by the Prophet Muhammad, peace be upon him. It didn't drop down from heaven as a printed text. Instead, it was revealed into the mind of the Prophet, peace be upon him. He recited it publicly, delivered it orally, recited it in the public prayers, and his companions then received a text. They memorized it and they passed it on to the next generation.
People who were not trained to read or write could still benefit from the Quranic revelation. They could know what God is saying to them and God is really saying it to them.
Yet, of course, through human mouthpieces.
Theoretically, God could have made it in writing and it would still have its own cadence in writing as well. Of course, it does even on the printed page, but it's not the need to make it poetic that caused it to be delivered orally. We can say that the fact that it will be delivered orally meant that God would make it interesting through that medium. One of the interesting features of the Quran is its rhyme, its rhythm, its cadence, and the beautiful sound of it.
Muslim tradition also has the idea that somehow the Quran was imprinted into the mind of the Prophet Muhammad, peace be upon him, not by being burnt into his mind because the mind is not a physical thing. Somehow the way it was revealed to the Prophet Muhammad, peace be upon him, it is not expected that he would forget it. At least that's the Muslim theology.
The Quran says in the 87th chapter, “We shall teach you the Quran and you shall not forget”.
Here, Allah says to the Prophet Muhammad, peace be upon him, that he will cause you to read in a way that you will not forget except what Allah wills. This means if Allah wills that the Prophet Muhammad, peace be upon him, should forget something, he would. Most likely, however, Allah does not want that the Prophet, peace be upon him, would forget anything. It's just that the Quran, again and again, emphasizes the absolute will of God. Whatever God wills is going to happen and whatever He wills not to happen cannot happen. In this case, because God is in a way depicting some power of the Prophet Muhammad, peace be upon him, he is not going to forget. The refrain is added, except what God wills. This is so that one does not get the idea that the Prophet Muhammad, peace be upon him, has a superhuman memory. One should always recall that God's will is what is overwhelming. He is empowering the Prophet Muhammad, peace be upon him, to memorize the revelation that's given to him.
It is obvious from the Quran itself that he had followers who were memorizing and learning the book. For example, in the 29th chapter of the Quran in the 46th verse, there is this discourse about the people doubting the Quran. Muslims are called upon to share the message of the Quran with other people. The responses of the other people are mentioned as well and some will object to the Quran. The Quran said that these are revelations in the hearts of those who have been given knowledge. That seems to point to the memorization process and the fact that it's not only the Prophet Muhammad, peace be upon him, who has memorized the texts, but there are people around him who are memorizing it as well. The 25th chapter of the Quran, again, gives the responses of the objectors. The Quran has them say that the Quran is ancient tales - “And they say, it is just ancient fables, which he has had written down: they are dictated to him morning and evening”. These are ancient tales that the Prophet Muhammad, according to the objectors, taught his followers and so they are then repeating it back to him. That might hint at the schooling process where a teacher reads to the students, the student tries to grasp the lesson and then reads it back to the master. This must have been what was happening in the lifetime of the Prophet Muhammad, peace be upon him, and the interchange of the Quran between him and his companions. As he is teaching them and they're reading it back to him.
Muslim tradition has a long list of names. Muhammad Mustafa Al-Azami was a great scholar from Riyadh, who is an Indian scholar, but he was most recently based in Riyadh where he taught and God have mercy on him. In his book, "The History of The Quran Text", he prepared a list of some 52 persons who have been known to memorize the Quran. He also wrote a book called "Kuttab al-Wahy” or “The Writers of the Revelation'' in which he has gone into more detail about the scribes. More specifically, the people who have been writing down the Quran in the time of the Prophet Muhammad, peace be upon him. Yet, such long lists come from piecing together many diverse reports and we cannot be confident that all of these reports are of the same quality in terms of their authenticity and verifiability. The hadith collection which is known as "Sahih Al-Bukhari" has some narratives which say that only four persons memorized the Quran in the time of the Prophet Muhammad, peace be upon him. One might expect that many more people had memorized at least portions of it and indeed many of the reports probably have stated that somebody was known to be a careful student of the Quran. They had memorized many parts of it but there was no clear testing process to say that one should sit down and recite the whole Quran and then be checked and certified. Many persons may have memorized large portions of the Quran, but four persons might have been certified by the Prophet, peace be upon him, himself because he heard them read the whole thing. He could testify that these are the persons that you can take the Quran from because they have passed their tests. This is probably why we have the mentioning that only four persons memorized the Quran in Bukhari. At the same time, we have diverse reports saying that other people memorized as well.
There is still value in memorizing the Quran today. At least one aspect of that value is not as prominent as it used to be in the past because writing materials were scarce, especially in the time of the Prophet, peace be upon him. Though the paper was invented in China, it would take another century for it to become widely used in the Prophet's community. So, memorizing the Quran at that time was a way of preserving it. Muslims could proudly boast and say that if their enemies took all of the copies of the Quran and burned them out of spite, the Muslims still have the Quran memorized in their minds and they could reproduce it by just gathering the readers together. However, now that the Quran has been preserved in writing, on memory sticks, and in the cloud, then that need for memorizing it is just not as prominent now. Nonetheless, Muslims see a great reward in memorizing the whole text. It still remains a matter of pride among Muslims that as the Quran itself says, God has made this book easy to memorize - is there anyone to memorize it? Muslims are responding to that challenge by memorizing the book.