The Qur'aan wasn't revealed at once. Its sentences were revealed on different occasions according to the appropriate time. A lot of the sentences were answers to questions, and it is uncommon to answer a question before the question is asked. For this reason, at the time of prophethood it was not possible to reveal the Qur'aan in a book form from the beginning and safeguard it. So in the early stages of Islam a lot of emphasis was given on memorizing. In the early stages of revelation, our prophet (s.a.w) used to repeat (the wordings) himself. So the words would go firmly into his memory. Regarding this, in one of the sentences of Surah Qiyaamah, Allah Ta'alah advised our prophet (s.a.w) that at the time of revelation you do not need to repeat the wordings quickly. Allah Ta'alah himself will create such a memory into our prophets mind that after the revelation, he will not be able to forget it. Therefore this is what happened that once the sentences were revealed, immediately he would memorize it. This way our Prophet's (s.a.w) heart was the most safe place for the Qur'aan in which there is absolutely no possibility for any minor mistake or changes. Then as a lesson for the Ummat and to show the blessings of the blessed month of Ramadhan, our Prophet (s.a.w) used to revise (read the Qur'aan thoroughly) to Hadhrat Jibrael (a.s) every Ramadhan. The year that our prophet passed away he revised the Qur'aan twice with Hadhrat Jibrael (a.s).
(Sahih Bukhari, Fathul Bari pg. 36, Vol I)
Then our prophet (s.a.w) taught his companions the meaning, he also taught the wordings, i.e. how they should be pronounced. The Sahabahs themselves had so much interest in learning the Qur'aan that in this subject they used to be worried about being in front of one another. Women didn't even ask from their husbands anything for dowry but for him to teach her the knowledge of the Qur'aan. Hundreds of Sahabahs gave there lives for nothing but this reason. Not only did they learn the Qur'aan but they revised it in the night IN SALAAH. Hadhrat Ubaadah bin Samit (r.a.) said, "Whenever a person migrated from Makkah to Madinah, our prophet (s.a.w) would give him to one of us Ansaarees to teach him the Qur'aan. In the Mosque of our prophet (s.a.w) the noise of learning and teaching of the Qur'aan used to reach such an extent that our prophet (s.a.w) had to remind them to lower their voices. So in the early stages of Islam greater emphasis was given on memorizing, and at that time this was the most safe and trustable way. There were very less people who could read or write, there were no publishers or press to publish books, therefore, if it was dependent upon writing then nor would they be able to publish so many Qur'aans, nor would it be so reliable, instead of that, Allah Ta'alah gave the Arabs such a powerful memory that each person could memorize thousands of poems and sayings. Even ordinary villagers not only knew their and their relatives family trees but they also knew their horses family trees. That is why this power of memorizing was used to safeguard the Qur'aan, and with this the paragraphs and chapters were spread to each and every corner of Arabia.