As Salaamu Alaikum!
Memorizing the Holy Qur’an is one of the most important ways to preserve Allah’s message. The others are to publish and distribute the book, or the text on the Internet and to recite the parts that one knows to other people. Prophet Muhammad (Sallallahu Alaihi Wassallam) said:
‘The best amongst you is the one who learns the Qur'an and teaches it.’ (Narrated by Al-Bukhari).
Actually, most important of all is to live by the message it contains, so as to be a good example to others. Aishah (may Allah be pleased with her), when asked to describe the character of Prophet Muhammad (Sallallahu Alaihi Wassallam) described it as ‘the Qur’an’.
The reward for remembering the message it contains, living by its guidance, and passing it on to others, is great:
"We, when the water (of Noah's Flood) overflowed beyond its limits, carried you, in the floating (Ark), That We might make it a Message unto you, and that ears (that should hear the tale and) retain its memory should bear its (lessons) in remembrance. "
Surah 69 Verses 11 - 12
It is Allah who helps people memorize the Qur’an, as he said to the Prophet Muhammad (Sallallahu Alaihi Wassallam)
By degrees shall We teach thee to declare (the Message), so thou shalt not forget,
Except as God wills: For He knoweth what is manifest and what is hidden.
And We will make it easy for thee (to follow) the simple (Path).
Therefore give admonition in case the admonition profits (the hearer).
Surah 87 Verses 6 - 9