Some logical arguments of this miracle
Even though majority Muslims believe this fact without the need of having proofs, nevertheless there are proofs that logically confirm this miracle and make our conviction stronger in its taking place. Sheikh Said Al Nursi lists them in his work Miracles of Prophet Muhammad, which makes sense of it’s happening:
The pagans of Makkah confirm it
Firstly this incident definitely took place because had it not taken place the disbelievers of Makkah would not have said that: “This is Magic!” as the Quran confirms it: “The Hour (of Judgement) is near, and the moon split. But if they see a Sign, they turn away, and say, 'This is transient magic.'” It is a fact that the disbelievers opposed Prophet Muhammad ﷺ and did everything in their will to eliminate him and his followers. Their claiming that it is a magic is sufficient enough to prove that they did indeed see it but they rejected and termed it as something else. The disbeliever’s claim that the Prophet ‘did magic’ is recorded in all biographies that have been ever written on his life, even in biographies written by Orientalist Western writers. Therefore, their claim of ‘his magic’ is accepted by all without any doubt!
Mutawatter (continuous) Tradition
Apart from the Quran this is recorded in the books of ahadith with no lesser classification than mutawatter. According to Ilm Al Hadith (Sciences of Hadith) a mutawatter hadith is reported by a larger number of narrators in every age, making it virtually impossible for any falsehood to have crept in to it. This classification is to be one of the most reliable. This splitting of the moon is a report which has been narrated by a great number of prophet's companions, and knowing how pious their lives were, it is almost impossible to discredit this report from their mouths as falsehood. Saad Al Din Taftazani (1322 – 1390 CE), a medieval Persian Islamic theologian said about this tradition that: “That is to say, these events had been passed down from group to group forming such a vast congregation that a conspiracy to lie would have been impossible. Like the appearance of the famous Haley's Comet a thousand years ago had been unanimously reported.”
Why did not the entire world see this event if it really took place?
Al Nursi gives a beautiful answer to this very logical question that arises in the mind of Non-Muslims and those Muslims who may have doubts: “Miracles take place in order to prove the claim of prophethood, to convince deniers, but not to force them to believe. This miracle was needed to convince those who had already heard of the prophethood of Muhammad ﷺ. To let it be seen from other parts of the world, or to demonstrate it in an undeniably obvious way, therefore, would have been contrary both to the wisdom of the All-Wise Creator and to the purpose of man's function in the universe, which is to open the way for the mind without disabling the free will. If the All-Wise Creator had, in accordance with the whims of materialist philosophers, left the moon in the same condition for a few hours to show it to the entire world, and if this fact had thereby been recorded by all historians, it would have been treated among other astronomical incidents, without being restricted to the messengership of Muhammad ﷺ and without being considered a proof of his prophethood. Or, it would have been such an obvious miracle that it would have forced the minds to believe and deprived it of the free will. As a consequence, coal and diamonds (or Abu Jahl (Prophets worst enemy) and Abu Bakr (Prophets best companion)) would have been the same -the purpose of man's function in the universe would have been lost. This may have been the reason why this miracle was not shown to the entire world to be recorded in all books of history.”
Why isn’t this recorded in the books of other cultures since it was astronomical incident?
To this the Al Nursi says “The mystery of guidance and the wisdom of the prophetic message required its demonstration to certain people defined by Allah's wisdom. Those who were not intended and who had not yet heard about the message of Muhammad ﷺ were prevented from seeing this occurrence by means of many obstacles such as clouds, fog, and time differences. Furthermore, and in addition to some other obstacles, it was sunset in Spain and England (which were then under the clouds of ignorance) when this incident happened, daytime in America, and morning in China and Japan; so it could not be observed in other parts of the world. Yet thoughtless rejecters say, "It did not occur, as it is not mentioned in the histories of such people as the Chinese, Japanese, and Americans.”
The biggest evidence of this miracle is the unanimous agreement of the entire Muslim world for the past fifteen centuries which is one of the beautiful tradition in Islam. Knowledge in Islam has been shared from generation to generation starting with the companions who learned directly from the Prophet ﷺ.
Sheikh says six evidences of its occurrence out of many:
Thus these major facts and most importantly, the unanimous historical agreement from earliest Islamic history with no difference among Islamic scholars, prove for us that Muhammad ﷺ did indeed split the moon, by the will of Allah ﷻ, as no human can do anything without the will of Allah ﷻ.
 Nursi, Bediuzzaman Said. 2010. Miracles of Prophet Muhammad - the miracle of splitting the moon Chapter 21 - Feedbooks
 Al Quran 54:1-2
 Also known as Al-Taftazani, was a great Muslim Persian scholar who had contributed towards fields of theology, Islamic jurisprudence, reason & philosophy, mathematics etc.
 Nursi 2010
 Nursi 2010