What is meant by inner rules in regards to recite the Holy Quran?

Since the Holy Quran is Allah's words and the Prophet of Islam's (P.B.U.H) miracle, Muslims show the special respect to it even at the advent of Islam. According to the verses of Quran, recommendations and narrations of the Holy Prophet (P.B.U.H), Muslims have been observing some special rules and terms in regards to recite the heavenly book that there hasn't had any previous record like that yet. This kind of reciting the Holy Quran, has become famous as apparent rules of reciting the Heavenly Book.
Since Quran, according to the traditions, has an exterior and some inners, its rules in regards to recite it has divided into the two rules: 1. Apparent rules 2. Inner (spiritual) rules.
This issue is mentioned by some verses of the Holy Quran and traditions of the Holy Prophet (P.B.U.H) and Imams (a.s).

Detailed Answer
It is said that the rules in regards to recite the Holy Quran can be divided into two divisions. The apparent rule and inner (spiritual) rule. However, it should be considered some rules and terms regarding the apparent rules which is mentioned in its place.
We can find the Quran inner rules by the Holy Quran itself and some traditions. The recommendation of the Holy Quran to those who want to recite it, is to recite it with Tarteel and think about what they are reciting profoundly. "Or a little more; and recite the Qur'an Slow and in a clear tone"[1] the Holy Quran said in another verse: "Those to whom We have sent The Scripture, and they recite it as it Should be recited, they believe [in The Qur'an] but those who disbelieve, They are indeed the losers."[2]
According to the traditions and Ahadiths, the two verses point to the both apparent and inner rules of the recitation of the Holy Quran. Imam Sadiq (a.s) said in regards to the verse 121 of the Sura Baqara: "we the Ahlul Bayt are the people who recite the Holy Quran in the right way (as it deserves to be recited)"[3]
Naturally, Imam Sadiq (a.s) declared the actual example of readers of the Holy Quran. It is likely that many believers who observe the both apparent and inner rules of the recitation of the Holy Quran can reach the level of real readers. Thus, Imam Sadiq (a.s) interpreted this verse "کِتابٌ أَنْزَلْناهُ إِلَیْکَ مُبارَکٌ لِیَدَّبَّرُوا آیاتِه"[4] that is as follows: "it means to recite the verses of the Holy Quran carefully, comprehend its truths and realities, do according to its rules and commends, be hopeful to its promises, be afraid of its warning, take an example of its stories, submit its orders and accept its prohibitions and interdictions. It doesn't mean to memorize the verses of the Holy Quran and recite it, because they memorize the Holy Quran's words but spoil its rights. It only means to ponder about verses of the Holy Quran and follow its rules and commands, as Allah (swt) says: "This Book which We have sent down to You [O, Messenger] is a Blessed Book [Of Wisdom] in order that the people Ponder about its Words of Revelation".[5]
According to this holy Hadith, we can point to some cases regarding the spiritual (inner) rules of recitation of the Holy Quran that are as follows:
1. Whenever they see a promise in the Holy Quran, they should be hopeful to reach it, and whenever see a warning, they should be afraid of being affected by that warning. Thus, if they find the heavenly or hellish verses of the Holy Quran, they are supposed to stay there and request Allah the Almighty to grant them the heaven and seek shelter in Allah from the devils' temptations.[6]
"They stand up on their own feet during the nights and recite the Holy Quran with the special rules and terms. Whenever they see the verses that contain encouragement and reward, they hope to obtain it and become eager that Allah the Almighty will grant them the reward, like those who expect to visit the one who they have been waiting for. And whenever they find the verses that contain intimidations and fears, they pay more attention to it, in a way such that their skins began to shudder and their heart dread of the fears. Their body become lean. It sounded like that the hellish exhalation and the voice of its flames and encountering its fiery chains are actually reverberating on their ears. So that they fall onto the ground and shed tears and seek shelter in Allah from the evils' temptations" Imam Ali (a.s) said regarding the features (characteristics) of the believers.[7]
2. What the other inner rules of the recitation of the Holy Quran is to ponder about its verses and comprehend them and then follow its rules and commands. Imam Sadiq (a.s) quoted from Imam Ali (a.s) that His Excellency said: "Be aware that you will get nothing if you recite the Holy Quran without pondering".[8]
3. Paying more attention that Quran is not the human's words and considering the Greatness of Allah that paying homage of His words is actually paying homage to Himself.
4. Tahliya is another inner rules of the recitation of the Holy Quran. Which means to coincide himself with the meaning of the verse he is reciting it. E.g. if he recites the story and tale of the Prophets (a.s), he should take it as an example or a lesson to learn. And if he recites Allah's names and qualities, he should ponder about their examples.[9]
5. Takhliyah is one of the rules of the recitation of the Holy Quran. Which means, when someone wants to learn something of Quran, in order to understand the Holy Quran better the one should remove all mental doubts from his mind.[10] This kind of rules called as Takhliyah.
6. Remove all undesirable qualities especially haughtiness, hypocrisy,[11] envy and greed is another inner rules in regards to recite the Holy Quran. So, if one couldn't remove them from himself, the meaning and concept of Allah's words won't manifest in his heart.
7. The spiritual purification is one of the highest of spiritual rules of the recitation of the Holy Quran. In other words, if one's heart is not pure, the one couldn't find the truth and reality of the Holy Quran as it says: "لا یمسّه الا المطهّرون" And that no one should touch it but The pure ones[12].
8. Seeing the Holy Quran as the sanative prescription is another spiritual rules of the Holy Quran that a reader of Quran should observe it, not as a simple text instead.
It is mentioned in a tradition that there are some people who see the Quran as A healing and seek medicine for their pains of it. So, Allah the Almighty, for the reason that those who take their cure of the Holy Quran, He keep pains and misfortunes away from people and sends down His Grace water on them.[13]

[1] . Muzammel, verse 4. "Aū zid ‘alaī‌hi warattilil‌qur‌āāna tar‌tīla"
[2] . Baqarah, verse 121, "Alladhêna ââtay‌nâhomol‌ketâba yat‌lônahô ḥaqqa telâwatehêê olââ‌eka yo‌menôna beh Wamay yak‌for behê fa‌ola‌eka homol‌khâserôn".
[3] . Kulaini, Muhammad Bn Yaghoub, Kafi, vol 1, p 215, Darul Kotob al-Islamiyah, Tehran, 1986.
[4] . Sura Sa'd, verse 29 "Ketâbon anzal‌nâhôô elay‌ka mobârakolleyaddabbarôô ââyâtehê".
[5] . Makarem Shirazi, Naser, Tafseer Nemooneh, vol 1, p 432, Darul-Kotob ul-Islamiya, Tehran, 1995.
[6] . Mashhadi, Muhammad, Tafseer Kanz ul-daghayegh wa Bahr ul-gharayeb, vol 2, p 132, Vezarat Irshad publication organization, Tehran, 1989.
[7] . Nahjul Balaghah, Sobhi Saleh, p 304.
[8] Kafi, vol 1, p 36.
[9] . Kashefi Sabzevari, Mulla Hussain, Jawaher al-tafseer, p 270, the office of Mirath Maktoob publication, Bita.
[10] . Ibid but with some changes.
[11] . Please refer to: Hur A'meli, Muhammad Bn Hassan, Wasai'l al-Shia, vol 6, p 182, A'lul Bayt (a.s), Qom 1409.
"إِنَّ مِنَ النَّاسِ مَنْ یَقْرَأُ الْقُرْآنَ لِیُقَالَ فُلَانٌ قَارِئٌ وَ مِنْهُمْ مَنْ یَقْرَأُ الْقُرْآنَ لِیَطْلُبَ بِهِ الدُّنْیَا وَ لَا خَیْرَ فِی ذَلِکَ وَ مِنْهُمْ مَنْ یَقْرَأُ الْقُرْآنَ لِیَنْتَفِعَ بِهِ فِی صَلَاتِهِ وَ لَیْلِهِ وَ نَهَارِهِ"
[12] . Waqi'ah, verse 79 "Lâ yamassohôô ellal‌moṭah‌harôn".
[13] . Kafee, vol 2, p 627.
" وَ رَجُلٌ قَرَأَ الْقُرْآنَ فَوَضَعَ دَوَاءَ الْقُرْآنِ عَلَى دَاءِ قَلْبِهِ فَأَسْهَرَ بِهِ لَیْلَهُ وَ أَظْمَأَ بِهِ نَهَارَهُ وَ قَامَ بِهِ فِی مَسَاجِدِهِ وَ تَجَافَى بِهِ عَنْ فِرَاشِهِ فَبِأُولَئِکَ یَدْفَعُ اللَّهُ الْعَزِیزُ الْجَبَّارُ الْبَلَاءَ وَ بِأُولَئِکَ یُدِیلُ اللَّهُ عَزَّ وَ جَلَّ مِنَ الْأَعْدَاءِ وَ بِأُولَئِکَ یُنَزِّلُ اللَّهُ عَزَّ وَ جَلَّ الْغَیْثَ مِنَ السَّمَاءِ فَوَ اللَّهِ لَهَؤُلَاءِ فِی قُرَّاءِ الْقُرْآنِ أَعَزُّ مِنَ الْکِبْرِیتِ الْأَحْمَرِ"