Abbas - Tanwîr al-Miqbâs min Tafsîr Ibn ‘Abbâs
(Our Lord!) O our Lord (And raise up in their midst) the offspring of Ishmael (a Messenger from among them) from their own lineage (who shall recite unto them Your revelations) the Qur'an, (and shall instruct them in the Scripture) the Qur'an (and in wisdom) the lawful and the prohibited (and shall make them pure) from sins through the declaration of the Oneness of God. (Lo! Thou, only Thou, art the Mighty) Who is vengeful against whosoever does not answer Your Messenger whom You sent to them, (the Wise) in sending the Messenger. And Allah answered his prayers and sent unto them Muhammad (pbuh); and these are the words with which Allah tried him, and he fulfilled them by asking Allah for them in his prayer.
Our Lord! And send among them, the people of this House, a messenger, one of them and God granted him this petition with [the sending of] Muhammad (s), who shall recite to them Your signs, the Qur’ān, and teach them the Book, the Qur’ān, and Wisdom, that is, what the former contains of judgments, and purify them, cleanse them of idolatry; You are the Mighty, the Victor, the Wise, in His creation.
Our Lord! And send among them a messenger, namely, Muḥammad, may God's blessings and peace be upon him, which is why he [Muḥammad] said: 'I am [the answer to] the supplication of my father Abraham, and the good tidings of Jesus and the vision of my mother - for she had seen in her sleep that a light was shining forth from her that illumined for her the palaces of Syria; [one of them, who shall recite to them Your signs, and teach them the Book and Wisdom, and purify them; You are the Mighty, the Wise.]
Ibrahim's Supplication that Allah sends the Prophet
Allah mentioned Ibrahim's supplication for the benefit of the people of the Sacred Area (to grant them security and provision), and it was perfected by invoking Allah to send a Messenger from his offspring. This accepted supplication, from Ibrahim, conformed with Allah's appointed destiny that Muhammad be sent as a Messenger among the Ummiyyin and to all non-Arabs, among the Jinns and mankind.
Hence, Ibrahim was the first person to mention the Prophet to the people. Ever since, Muhammad was known to the people, until the last Prophet was sent among the Children of Israel, Jesus the son of Mary, who mentioned Muhammad by name. Jesus addressed the Children of Israel saying,
(My mother saw a light that went out of her and radiated the palaces of Ash-Sham.)
It was said that the Prophet's mother saw this vision when she was pregnant with, narrated this vision to her people, and the story became popular among them. The light mentioned in the Hadith appeared in Ash-Sham (Greater Syria), testifying to what will later occur when the Prophet's religion will be firmly established in Ash-Sham area. This is why by the end of time, Ash-Sham will be a refuge for Islam and its people. Also, Jesus the son of Mary will descend in Ash-Sham, next to the eastern white minaret in Damascus. The Two Sahihs stated,
(There will always be a group of my Ummah who will be on the truth, undeterred by those who fail or oppose them, until the command of Allah comes while they are on this.)
Al-Bukhari added in his Sahih, (And they will reside in Ash-Sham.)
The Meaning of Al-Kitab wal-Hikmah
(and instruct them in the Book) meaning, Al-Qur'an,
(and Al-Hikmah) meaning, the Sunnah, as Al-Hasan, Qatadah, Muqatil bin Hayyan and Abu Malik asserted. It was also said that `Al-Hikmah', means `comprehension in the religion', and both meanings are correct. `Ali bin Abi Talhah said, that Ibn `Abbas said that,
(and purify them) means, "With the obedience of Allah.''
إِنَّكَ أَنتَ العَزِيزُ الحَكِيمُ
(Verily, You are the Mighty, the Wise).
This Ayah stated that Allah is able to do anything, and nothing escapes His ability. He is Wise in His decisions, His actions, and He puts everything in its rightful place due to His perfect knowledge, wisdom and justice.
Maududi - Sayyid Abul Ala Maududi - Tafhim al-Qur'an
"Purify their lives:" purify their beliefs, ideas, deeds, habits, customs, culture, politics, in short, every aspect of life.
As.Allah has full powers and perfect wisdom, He granted this prayer of the Prophet Abraham and made Hadrat Muhammad (Allah's peace be upon him) His Messenger.
Our Lord! And send among them a messenger one of them who shall recite to them Your signs and teach them the Book and wisdom and purify them; You are the Mighty the Wise. Because the obligations were received from messengers rather than being understood by reason alone [Abraham] asked that they not be left to themselves without a messenger or law. He requested that the messenger be one of them so that they would feel more at home with him and it would be easier for them. Its meaning rightly understood is that when He سبحانه informed [Abraham] of the state of our Prophet ﷺ [Abraham] asked for the completion of what He had promised by way of the one He commanded.