"When the Messenger of Allah (may Allah's peace and blessings be upon him) got up to pray at night, he would make "Takbīr"." In other words, he would begin his prayer with the statement: "Allah is the Greatest." Takbīr is a pillar without which prayer is not valid.
"And then say: 'Glory be to You, O Allah.'" Meaning: I declare You above and free from all shortcomings and imperfections and from all that does not befit You and Your majesty, my Lord.
"And praise", meaning: praising Allah and thanking him for this help. In other words: If it were not for Your help and guidance, I would not be glorifying You.
So this is an acknowledgment from the slave of the favors bestowed by his Lord as well as an acknowledgment of his own helplessness without the help of Allah, the Almighty.
"Blessed is Your name." Your name is ever abundant, perfect, and broad, and its blessings are widely spreading in the heavens and the earth. This emphasizes that all good lies in the hand of Allah, the Almighty.
"Exalted is Your greatness."
Your greatness rises, elevates, and exalts over all greatness; Your sublimity is high above all others; and Your authority subdues all others. The greatness of Allah is far elevated above having a partner in His dominion, His lordship, His divinity, or His names and attributes.
That is why he said thereafter: "There is no god but You", meaning: there is no deity truly worthy of being worshiped except You. You are the Only One deserving to be worshiped, alone without any partner; You, the One Who possesses the most praiseworthy attributes and bestows the most immense favors.
In this opening invocation, we praise Allah, the Almighty, and declare Him above and free from all shortcomings and imperfections and from all that does not befit Him and His majesty.
This is one of the various invocations reported in this respect. It is better to say this and that from time to time, thus observing all authentic aspects of the Sunnah and not disregarding any of them.
In one of those various wordings, the Prophet would reportedly say: "There is no god but Allah" three times. This statement means: There is no deity truly worthy of being worshiped except You.
In the Qur’an, Allah, the Almighty, says: {That is because Allah is the Truth, and that which they call upon other than Him is falsehood.} {Sūrat al-Hajj: 62]
"Thereafter, he would say: 'Allah is the absolute Greatest,' three times." He, Glorified and Exalted, is Greater than all things.
Then, after opening his prayer, he would seek refuge in Allah from the accursed Devil, saying: "I seek refuge in Allah, the All-Hearing and All-Knowing."
In other words, I seek the refuge, support, and protection of the All-Hearing and All-Knowing Lord.
"From the accursed", meaning: the outcast, rejected, and expelled from the mercy of Allah.
"Devil", meaning: the rebellious and insolent devils from among the Jinn and mankind. O Allah, Do not give him authority whereby he can harm me in my religious and worldly affairs or prevent me from doing such things that will benefit me in my religious and worldly affairs. Whoever seeks refuge in Allah has indeed turned to the might and power of the Creator against their enemies who want to cut them off from their Lord and lead them into the deep pits of evil and destruction.
"From his evil suggestions", meaning: madness and epilepsy that afflict mankind.
Since Satan may touch a person with madness, it was prescribed by the Shariah to seek refuge against him.
"His blowing", meaning: pride. Through insinuations, Satan blows into a man and prompts him to see himself big and others little. Thus, his pride and arrogance grows.
"His spitting", meaning: witchcraft. This is the evil of magicians, who tie the strings and then blow on each knot so as to produce the effect intended by magic.
"After that, he would recite." He would recite Qur’an, opening with Sūrat al-Fātihah.
Who passed it on, from Imam an-Nasa'i back to the one who was there.