I am afraid I will not be able to present the explanations offered by all the authorities on the issue. I will restrict myself to the most plausible and well-argued stance in this regard.
A simple translation of the verse is produced below:
He (Solomon) said: O courtiers! Which of you will bring me her throne before they present themselves humbly before me? An Ifrît (strong) from the Jinn said: I will bring it before you before you rise from your place. Lo! I am indeed strong and trusty for such work.
The man (among the courtiers) who possessed the knowledge of the Book said: I will bring it to you within the twinkling of an eye. And [there it was]. When he (Solomon) saw it set before him, he said: This is boundless bounty of my Lord, that [has been showered upon me] so that He may try me whether I offer thanks or prove ungrateful. Whosoever shows thankfulness only does so for his own benefit: and whosoever is ungrateful [harms his own interest]. For! My Lord is in Independent, Bountiful. [38:40]
The Holy Qur’an does not mention the method applied for transferring the throne of the queen. One can only make mere conjecture that must be avoided. However, the Qur’an has alluded to the fact that the fountainhead of the knowledge whim was used by the men was the divine guidance contained in the scripture. The very mention that he possessed knowledge of the scripture clearly indicates to this. It seems some people had learnt this kind of knowledge. At another place in the Holy Qur’an it is told that two angels were sent on the earth who taught this kind of supernatural knowledge. This knowledge was to be used very cautiously. The angels [see: 2:102] would clearly indicate that it was a sort of test and that the acquired knowledge could used both for the positive and negative purposes. This indicates that there existed the knowledge of such powers. However, one cannot explain the mode of operation in this case.
As regards the matter of Law-e-Mahfooz, فِي لَوْحٍ مَّحْفُوظٍ literally a guarded tablet, I would like to mention two things. First, the Holy Qur’an fully conveys the message employed and there is no need of splitting hair regarding the nature of the tablet. If you go through the Surah you will find no problem in ascertaining the meaning implied. The Almighty says that the Qur’an is something to be taken seriously. It is the divine word which is not accessed at by the satans and jinns so that they cannot affect its integrity. The same subject has been taken in various other places in the Qur’an.
Some scholars even tend to find the nature of the tablet (Lawh). Some think it is places upon the seventh heaven and there are some traditions ascribed to the companions and the prophet which give details of this tablet. But we believe that is not the basic purpose of the Qur’an to lead people to strive for discovering what has no relevance with their beliefs and practices. An analogy would best explain my point. God says that he has created man with His hands. The purpose of the mentioning the fact is clear. There are those who receive the message and act accordingly. While others strive to deduce that the verse actually indicates to the fact that the Almighty has two hands like that of ours, or bigger or smaller according to their own assumed scales. This attitude is totally against the Qur’anic purpose. We believe that there may or may not be a tablet places somewhere in the heavens but that is not our concern. The Qur’an does not want us to consider the nature of mentioned things rather it communicates other facts which are well taken by those who intend to receive guidance from it.
 And follow that which the devils falsely related against the kingdom of Solomon. Solomon disbelieved not; but the devils disbelieved, teaching mankind magic and that which was revealed to the two angels in Babel, Harut and Marut. Nor did they (the two angels) teach it to anyone till they had said: We are only a (fitnah) test, therefore disbelieve not (in the guidance of Allah).