What Is the Meaning of the Basmalah?
Question

What is the meaning of the Basmalah [the Arabic words Bismillah al-Rahman al-Rahim (In the name of Allah, the Most Gracious, the Most Merciful)]? 
And what is meant by the words “Iqra bismi Rabbika” (Read (or recite) in the name of your Lord – [al-‘Alaq 96:1 – interpretation of the meaning])?
Praise be to Allah.What does Bismillah mean?
When one says “Bismillah ” when starting to do anything, what that means is, “I start this action accompanied by the name of Allah or seeking help through the name of Allah, seeking blessing thereby. Allah is God, the beloved and worshipped, to Whom hearts turn in love, veneration and obedience (worship). He is al-Rahman (the Most Gracious) Whose attribute is vast mercy; and al-Rahim (the Most Merciful) Who causes that mercy to reach His creation. 
It was said that what this means is: I start this action by mentioning the name of Allah. 
Ibn Jarir (may Allah have mercy on him) said: 
“Allah, may He be exalted and His name sanctified, taught His Prophet Muhammad (peace and blessings of Allah be upon him) proper manners by teaching him to mention His most beautiful names before all his actions. He commanded him to mention these attributes before starting to do anything, and made what He taught him a way for all people to follow before starting anything, words to be written at the beginning of their letters and books. The apparent meaning of these words indicates exactly what is meant by them, and it does not need to be spelled out.” 
There is something omitted in the phrase “Bismillah” when it said before starting to do something, which may be “I begin my action in the name of Allah,” such as saying, “In the name of Allah I read”, “In the name of Allah I write”, “In the name of Allah I ride”, and so on. Or, “My starting is in the name of Allah”, “My riding is in the name of Allah”, “My reading is in the name of Allah”, and so on.  It may be that blessing comes by saying the name of Allah first, and that also conveys the meaning of starting only in the name of Allah and not in the name of anyone else. 
The name of Allah is the greatest name and is so well known as to need no explanation; this is a name that belongs exclusively to the Creator and no one else. The correct view is that it is derived from the root ilah. He is God (ilah) which means that He is worshipped and is divine. 
Al-Rahman is one of the names of Allah that belong exclusively to Him. It means the One Who possesses vast mercy, because this form (fa’lan) is indicative of fullness and abundance. It is the most exclusive name of Allah after His name Allah, just as mercy is His most exclusive attribute. Hence this name (al-Rahman) often appears after the name Allah, as in the ayah (interpretation of the meaning): 
“Say (O Muhammad): Invoke Allah or invoke the Most Gracious [al-Rahman] (Allah)” [al-Isra 17:110]
Al-Rahim is also one of the names of Allah, and means the One Who causes His mercy to reach those whom He wills among His slaves.  
Ibn al-Qayyim (may Allah have mercy on him) said: 
“Al-Rahman refers to an attribute that is connected to Allah and is part of His Essence, and al-Rahim refers to a connection with the one to whom mercy is shown. The former is adjectival (referring to what He is) and the latter is verbal (referring to what He does). The former indicates that mercy is His attribute, and the latter indicates that He bestows His mercy upon His creation. If you want to understand this then ponder the meaning of these verses (interpretation of the meanings): 
“And He is Ever Most Merciful (Rahim) to the believers”[al-Ahzab 33:43]  
“Certainly, He is unto them full of kindness, Most Merciful (Rahim)” [al-Tawbah 9:117]
The word al-Rahman is never used in this context. So we know that the word Rahman means the One Whose attribute is mercy (rahmah), and al-Rahim is the One Who bestows His mercy.” (Badai’ al-Fawaid, 1/24) 
Saying the Basmalah when reading the Quran
The ruling on saying the Basmalah before reading Quran depends on the situation: 

If it is at the beginning of a surah – apart from Surat Bara-ah (al-Tawbah) – then the majority of imams have stated that “it is mustahabb to recite the Basmalah at the beginning of each surah, in prayer or otherwise.

This should be done as a regular practice, and some of them considered that a reading of the whole Quran is incomplete if the Basmalah was not recited at the beginning of every surah apart from Bara-ah (al-Tawbah).” When Imam Ahmad (may Allah have mercy on him) was asked about reciting it at the beginning of every surah, he said, “Do not neglect it.” 

If one is starting in the middle of a surah – which is the case asked about in the question – then the majority of scholars and Quran readers say that there is no reason why one should not start with it. 

It was said to Imam Ahmad, after he had said that it should not be omitted at the beginning of the surah, “What if a person starts reading part way through a surah?” He said, “There is nothing wrong [with saying the Basmalah].” Al-Abadi narrated that al-Shafi’i (may Allah have mercy on him) regarded it as mustahabb (to say the Basmalah, when starting to recite) part way through a surah.
The Quran readers said: It is certain that one should say the Basmalah if the ayah which will be read after saying it contains a pronoun that refers to Allah, such as the verses (interpretation of the meanings): 
“To Him (Alone) is referred the knowledge of the Hour” [Fussilat 41:47] 
“And it is He Who produces gardens” [al-An’am 6:141] because otherwise, if one recites these verses after seeking refuge with Allah from the Shaytan, the pronoun may appear to refer to the Shaytan which would convey an abhorrent meaning. 

Reciting the Basmalah at the beginning of Surat Bara-ah (al-Tawbah); there is hardly any dispute among the scholars that doing this is makruh (disliked). 

Salih said concerning some issues that he narrated from his father Ahmad (may Allah have mercy on him): “I asked him about Surah al-Anfal and Surat al-Tawbah, whether it is permissible for a man to separate them by saying Bismillah al-Rahman al-Rahim. My father said: ‘With regard to the Quran, reference should be made to what the Companions of the Messenger of Allah (peace and blessings of Allah be upon him) were agreed on; nothing should be added to or taken away from that.’”  

Reciting it part way through Surat al-Bara-ah (al-Tawbah) . The Quran readers differed concerning that, as was narrated by Ibn Hajar al-Haythami in al-Fatawa al-Fiqhiyyah (1/52), and he said: “Among the leading Quran readers, al-Sakhawi said that there is no dispute that it is Sunnah to start with the Basmalah when one starts reading part way through this surah [al-Tawbah], as he differentiated between starting at the beginning and starting in the middle, but his explanation was facile and was refuted by al-Ja’bari from among the Quran readers. This is more likely (i.e., the view that it is makruh is more likely to be correct), because the reason why the Basmalah should not be recited at the beginning (of al-Tawbah) is that it came with the sword (i.e., the command to fight the kuffar) and it exposes the hypocrites and their foul deeds in a manner that is not unlike any other surah, and this theme is repeated throughout Surat al-Tawbah. Therefore it is not prescribed to recite the Basmalah even if one starts reciting part way through this surah, just as it is not prescribed at the beginning, for the reasons we have established.” 

See al-Adab al-Shar’iyyah by Ibn Muflih, 2/325; al-Mawsu’ah al-Fiqhiyyah, 13/253; al-Fatawa al-Fiqhiyyah al-Kubra, 1/52 
Meaning of ”Iqra bismi Rabbika”
With regard to the meaning of the words, “Iqra bismi Rabbika” (Read (or recite) in the name of your Lord – [al-‘Alaq 96:1 – interpretation of the meaning]), Imam Ibn Jarir (may Allah have mercy on him) said: “The interpretation of the words ‘Iqra bismi Rabbika’ is that they were addressed to Muhammad (peace and blessings of Allah be upon him), in other words, Read, O Muhammad, by mentioning the name of your Lord (Who created).” 
And Allah knows best.