There are several hadiths (sayings of the Prophet Muhammad, peace be upon him) that highlight the importance and virtues of becoming a hafiz (one who has memorized the Quran).
Prophet (PBUH) said
"The Quran will come on the Day of Resurrection and will say, 'O Lord, adorn him.' So he will be given a crown of honor to wear. Then it will say, 'O Lord, give him more.' So he will be given a garment of honor.' Then it will say, 'O Lord, be pleased with him.' So Allah will be pleased with him." (Narrated by At-Tirmidhi)
The Messenger of Allah (PBUH) said:
“Allah has angels who roam the roads looking for those who mention Allah. When they find some people who mention Allah, they call each other, saying, ‘Come to the object of your pursuit.’ “He [Prophet] added, ‘Then the angels encircle them with their wings up to the sky of the world…'” [Al-Bukhari]