Hadith: The Messenger of Allah (may Allah's peace and blessings be upon him) taught me the Tashahhud, with my hand between his two hands, as he would teach me a Surah from the Qur’an

عَنِ ابْنَ مَسْعُودٍ رضي الله عنه قَالَ:
عَلَّمَنِي رَسُولُ اللَّهِ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ، وَكَفِّي بَيْنَ كَفَّيْهِ، التَّشَهُّدَ، كَمَا يُعَلِّمُنِي السُّورَةَ مِنَ القُرْآنِ: «التَّحِيَّاتُ لِلَّهِ، وَالصَّلَوَاتُ وَالطَّيِّبَاتُ، السَّلاَمُ عَلَيْكَ أَيُّهَا النَّبِيُّ وَرَحْمَةُ اللَّهِ وَبَرَكَاتُهُ، السَّلاَمُ عَلَيْنَا وَعَلَى عِبَادِ اللَّهِ الصَّالِحِينَ، أَشْهَدُ أَنْ لاَ إِلَهَ إِلَّا اللَّهُ، وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ». وفي لفظ لهما: «إِنَّ اللهَ هُوَ السَّلَامُ، فَإِذَا قَعَدَ أَحَدُكُمْ فِي الصَّلَاةِ فَلْيَقُلْ: التَّحِيَّاتُ لِلَّهِ وَالصَّلَوَاتُ وَالطَّيِّبَاتُ السَّلَامُ عَلَيْكَ أَيُّهَا النَّبِيُّ وَرَحْمَةُ اللهِ وَبَرَكَاتُهُ، السَّلَامُ عَلَيْنَا وَعَلَى عِبَادِ اللهِ الصَّالِحِينَ، فَإِذَا قَالَهَا أَصَابَتْ كُلَّ عَبْدٍ لِلَّهِ صَالِحٍ فِي السَّمَاءِ وَالْأَرْضِ، أَشْهَدُ أَنْ لَا إِلَهَ إِلَّا اللهُ، وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ، ثُمَّ يَتَخَيَّرُ مِنَ الْمَسْأَلَةِ مَا شَاءَ».
[صحيح] - [متفق عليه] - [صحيح البخاري: 6265]

Ibn Mas‘ūd (may Allah be pleased with him) reported:
The Messenger of Allah (may Allah's peace and blessings be upon him) taught me the Tashahhud, with my hand between his two hands, as he would teach me a Surah from the Qur’an: "At-tahiyyātu lillāhi was-salawātu wat-tayyibātu as-Salāmu ‘alayka ayyuha an-nabiyyu wa rahmatullāhi wa barakātuh, as-salāmu ‘alayna wa ‘ala ‘ibādillāhi as-sālihīn, ash-hadu alla ilāha illallāh wa ash-hadu anna Muhammadan ‘abduhu wa rasūluh (Greetings, prayers, and good things are due to Allah. May the peace, mercy, and blessings of Allah be upon you, O Prophet. Peace be upon us and upon the righteous slaves of Allah. I bear witness that there is no god but Allah, and I bear witness that Muhammad is His slave and Messenger)." Another wording reads: "Verily, Allah is the Giver of Peace, so when any of you sits in prayer, let him say: 'At-tahiyyātu lillāhi was-salawātu wat-tayyibātu as-salāmu ‘alayka ayyuha an-nabiyyu wa rahmatullāhi wa barakātuh, as-salāmu ‘alayna wa ‘ala ‘ibādillāhi as-sālihīn (Greetings, prayers, and good things are due to Allah. May the peace, mercy, and blessings of Allah be upon you, O Prophet. Peace be upon us and upon the righteous slaves of Allah).' If you say this, it will benefit every righteous slave of Allah in heaven and on earth. (Then, say:) 'Ash-hadu alla ilāha illallāh wa ash-hadu anna Muhammadan ‘abduhu wa rasūluh (I bear witness that there is no god but Allah, and I bear witness that Muhammad is His slave and Messenger).' Then, let him choose whatever supplication he would like."
[Authentic hadith] - [Narrated by Bukhari & Muslim] - [Sahih Bukhari - 6265]

Explanation

The Prophet (may Allah's peace and blessings be upon him) taught Ibn Mas‘ūd (may Allah be pleased with him) the Tashahhud, which is to be recited in prayer, while holding his hand between both his hands to attract his attention to him, just as he would teach him a Surah from the Qur’an. This signifies how important the Prophet regarded this Tashahhud, its words and meanings. The Prophet (may Allah's peace and blessings be upon him) said: "At-tahiyyātu lillāh": It means that every word or deed denoting glorification is due to Allah Almighty. "As-salawāt": It means that the known prayers, both the obligatory and the voluntary, are due to Allah Almighty. "At-tayyibāt": It means that the good words, deeds, and attributes that denote perfection are all due to Allah Almighty. "As-salāmu ‘alayka ayyuha an-nabiyyu wa rahmatullāhi wa barakātuh": A supplication for him to be granted safety from every evil and harm and to be granted more of all good things. "As-salāmu ‘alayna wa ‘ala ‘ibādillāhi as-sālihīn": It is a supplication for the one praying and for every righteous slave in heaven and on earth to be granted safety. "Ash-hadu alla ilāha illallāh": It means I firmly acknowledge that there is no true god but Allah. "Wa anna Muhammadan ‘abduhu wa rasūluh": I affirm that he is a slave of Allah and His last Messenger.
Then, the Prophet (may Allah's peace and blessing be upon him) urged the one praying to choose whatever supplication he would like.