There are lots of reasons why Muslims should memorize Quran
The Prohohet Muhammed (PBUH) said
خَيْرُكُمْ مَنْ تَعَلَّمَ الْقُرْآنَ وَ عَلَّمَه
It means that "the best of us are those who learn Quran and teaches it."
The Prophet (PBUH) also said
"It will be said to the companion of the Qur'aan after he has entered Paradise, 'Recite, and rise!' For every verse he recites he will rise one level (in Paradise), until he recites the last verse with him (i.e., in his memory)."
“Whoever recites one Word from the Book of Allah will be rewarded for a good deed and ten more like it; and I don’t say that Alif Lam Mim is a letter but Alif is a letter and Lam is a letter and Mim is a letter”