Allaah سبحانه و تعالى says,
Imaam Ibn al-Qayyim رحمه الله has given the wisdom behind seeking refuge in Allaah from Shaytaan before reading the Qur'aan:
(1) The Qur'aan is a healing for whatever is in the heart; it removes whatever comes from Shaytaan, be it his evil whisperings, lusts or evil desires. It is the antidote to what Shaytaan has put into the heart. So first the person is asked to reject whatever is in his heart from the acts of Shaytaan. Therefore, the medicine will find the heart free and can take its proper place and have its proper effect. As is said in poetry, "Her desire came to me before I knew the desire/ And it met my heart empty so it took root therein." So this healthy medicine comes to the heart and cleanses it from what can damage it. And it becomes successful therein.
(2) The Qur'aan is the source of guidance, knowledge and good in the heart of the reader, in the same way that water is the source that brings forth plants. Shaytaan is a fire that burns down the plants, one by one. For every good plant that springs in the heart, Shaytaan attempts to destroy it and burn it down. And Allaah سبحانه و تعالى orders us to seek refuge in Him in order for Him to ruin what Shaytaan is attempting to do. The first point stated above deals with the benefits of the Qur'aan while this point actually deals with the preservation and remaining of the Qur'aan in the heart.
(3) The angels descend upon the one who reads the Qur'aan and listen to its recitation. This has been related in a hadeeth in which Usayd Ibn Hudayr رضي الله عنه was reciting the Qur'aan and he saw a canopy with lamps in it. The Prophet صلى الله عليه وسلم told him that those were angels. And the devils are the opposite of the angels and are their enemies. The one who reads the Qur'aan has been ordered to ask Allaah سبحانه و تعالى that he should be put far away from his enemy, the devils, until he is in the midst of only angels. This situation cannot possibly be occupied by both the the angels and the devils at the same time.
(4) Shaytaan tries to disturb the one who reads the Qur'aan with his steeds and feet until the reader is void of the meanings of the Qur'aan. The reader should try to ponder over and understand the words of Allaah سبحانه و تعالى. Shaytaan tries to come between his heart and the meanings of the Qur'aan. Thus the reader is not able to benefit from the Qur'aan. Therefore we have been ordered to seek refuge from this Shaytaan when we are about to read the Qur'aan.
(5) The reader of the Qur'aan is in a private conversation with Allaah سبحانه و تعالى. And Allaah سبحانه و تعالى listens closer to His words when recited in a pleasant way than a singer listens to her song. The song of Shaytaan is poetry and music. The reader has been ordered to seek refuge from Shaytaan so Shaytaan will be removed from the private conversation and Allaah سبحانه و تعالى will isten to the person's recital.
(6) Allaah سبحانه و تعالى has stated that whenever He sent a Messenger or a Prophet, Shaytaan has always tried to interfere in what he is preaching by bringing words or ideas that do not originate with Allaah سبحانه و تعالى. The early scholars all agreed that this meant that when one is reciting, Shaytaan throws some words or such into the recital. If this was done with respect to the Prophets عليهم الصلاة والسلام, what must be the case with respect to others? This is why the reciter will sometimes make mistakes. And why he sometimes gets confused and cannot recite properly. Or his mind and his heart becomes confused. When he begins to recite, he might find one of these, if not all of them, occurring.
(7) Whenever a person contemplates the doing of a good deed, Shaytaan becomes even more zealous in his attempts to come between the person and the good deed, to prevent the person from doing the good deed. [Therefore, we are asked to seek refuge in Allaah سبحانه و تعالى in order for Shaytaan to be prevented from stopping us from this good deed.] 
Shaykhul-Islaam Ibn Taymiyyah رحمه الله elaborated on that last point when he said that one who prays has to face more of Shaytaan's whisperings than one who does not pray. The same is true for those who are students of knowledge. This is because they are greater enemies to Shaytaan and he makes his attack on them more fierce. One of the moments in which Shaytaan is most likely to attack humans is when they try to read the Qur'aan. It is this moment when the person is in most need of seeking Allaah's protection سبحانه و تعالى from Shaytaan so that he will be able to benefit from his reading of the Qur'aan. 
Finally, Imaam al-Haafidh Ibn Katheer رحمه الله notes that when seeking refuge in Allaah سبحانه و تعالى from Shaytaan - in general, not just upon reading the Qur'aan - one recognises his own inability to defeat that hostile but hidden enemy. That enemy cannot be repelled like evil humans by behaving kindly toward them or by some type of "bribe." Instead, one must turn only to Allaah سبحانه و تعالى to repel that enemy and allow the person to freely read and benefit from the Qur'aan.