What Does It Mean to Recite Sūrat al-Tawḥīd in the Hadiths?

It is better to understand the hadith in this way:
عَنْ عَبْدِ اللَّهِ بْنِ حَيٍّ قَالَ سَمِعْتُ أَمِيرَ الْمُؤْمِنِينَ ع يَقُولُ مَنْ قَرَأَ قُلْ هُوَ اللَّهُ أَحَدٌ إِحْدَى عَشْرَةَ مَرَّةً فِي دُبُرِ الْفَجْرِ لَمْ يَتْبَعْهُ فِي ذَلِكَ الْيَوْمِ ذَنْبٌ وَ إِنْ رَغِمَ أَنْفُ الشَّيْطَانِ.
*Thawāb al-aʿmāl wa ʿiqāb al-aʿmāl*, text, p. 129

ʿAbd Allāh ibn Ḥayy relates: I heard the Commander of the Faithful, peace be upon him, say, “Whoever recites ‘Say: He is God, the One’ eleven times after the dawn prayer will not be pursued by any sin that day, even if Satan’s nose is rubbed in the dust” (that is, Satan despairs of overcoming him).

Assuming the hadith is reliably transmitted, reciting Sūrat al-Tawḥīd is, in reality, an affirmation and response to the testimony of the Most High Lord to His oneness. The necessary consequence of this affirmation is tawḥīd—“tawḥīd in the true sense” means “sincerity in affirming God’s oneness.” Thus, when a person who is devoted to tawḥīd recites this blessed sūrah in the manner described, no sin can draw near to him, and Satan’s efforts to deceive him will come to nothing.

Recitation rooted in tawḥīd is the hardware and software of Sūrat al-Tawḥīd, and it brings its own particular effects and blessings.

God knows best…

I offer you some other hadiths along the same lines.

َ قَالَ أَبُو عَبْدِ اللَّهِ ع مَنْ قَرَأَ قُلْ هُوَ اللَّهُ أَحَدٌ حِينَ يَخْرُجُ مِنْ مَنْزِلِهِ عَشْرَ مَرَّاتٍ لَمْ يَزَلْ فِي حِفْظِ اللَّهِ عَزَّ وَ جَلَّ وَ كِلَاءَتِهِ حَتَّى يَرْجِعَ إِلَى مَنْزِلِهِ
*Al-Kāfī* (Islamic edition), vol. 2, p. 542

Whoever recites “Say: He is God” ten times when leaving his home will remain under God’s protection until he returns home.

َ عَنْ أَبِي جَعْفَرٍ ع قَالَ: مَنْ قَرَأَ قُلْ هُوَ اللَّهُ أَحَدٌ مَرَّةً بُورِكَ عَلَيْهِ وَ مَنْ قَرَأَهَا مَرَّتَيْنِ بُورِكَ عَلَيْهِ وَ عَلَى أَهْلِهِ وَ مَنْ قَرَأَهَا ثَلَاثَ مَرَّاتٍ بُورِكَ عَلَيْهِ وَ عَلَى أَهْلِهِ وَ عَلَى جِيرَانِهِ وَ مَنْ قَرَأَهَا اثْنَتَيْ عَشْرَةَ مَرَّةً بَنَى اللَّهُ لَهُ اثْنَيْ عَشَرَ قَصْراً فِي الْجَنَّةِ فَيَقُولُ الْحَفَظَةُ اذْهَبُوا بِنَا إِلَى قُصُورِ أَخِينَا فُلَانٍ فَنَنْظُرَ إِلَيْهَا وَ مَنْ قَرَأَهَا مِائَةَ مَرَّةٍ غُفِرَتْ لَهُ ذُنُوبُ خَمْسٍ وَ عِشْرِينَ سَنَةً مَا خَلَا الدِّمَاءَ وَ الْأَمْوَالَ وَ مَنْ قَرَأَهَا أَرْبَعَمِائَةِ مَرَّةٍ كَانَ لَهُ أَجْرُ أَرْبَعِمِائَةِ شَهِيدٍ- كُلُّهُمْ قَدْ عُقِرَ جَوَادُهُ وَ أُرِيقَ دَمُهُ وَ مَنْ قَرَأَهَا أَلْفَ مَرَّةٍ فِي يَوْمٍ وَ لَيْلَةٍ لَمْ يَمُتْ حَتَّى يَرَى مَقْعَدَهُ فِي الْجَنَّةِ أَوْ يُرَى لَهُ.
*Al-Kāfī* (Islamic edition), vol. 2, p. 619

It is related from Imam Muḥammad al-Bāqir, peace be upon them both, that he said: “Whoever recites the sūrah ‘قُلْ هُوَ اللَّهُ أَحَدٌ’ will be blessed and attain an abundance of goodness. Whoever recites it twice will bring blessing upon himself and his family; whoever recites it three times will bring blessing upon himself, his family and his neighbours; and whoever recites it twelve times, God will build for him twelve palaces in Paradise. Then the scribes who record deeds will say to one another, ‘Come, let us go to the palaces of our brother so-and-so and look upon them.’ Whoever recites it a hundred times will have the sins of twenty-five years forgiven, except for wrongful bloodshed and taking other people’s property. Whoever recites it four hundred times will receive the reward of four hundred martyrs, each of whom had his horse hamstrung and his blood shed. And whoever recites it a thousand times in a day or a night will not die before seeing his place in Paradise, or having it shown to him”—that is, someone will see it in a dream.

َ قَالَ رَسُولُ اللَّهِ ص مَنْ قَرَأَ قُلْ هُوَ اللَّهُ أَحَدٌ مِائَةَ مَرَّةٍ حِينَ يَأْخُذُ مَضْجَعَهُ غَفَرَ اللَّهُ لَهُ ذُنُوبَ خَمْسِينَ سَنَةً.
*Al-Kāfī* (Islamic edition), vol. 2, p. 620

The Messenger of God, peace and blessings be upon him and his family, said: “Whoever recites the sūrah ‘Say: He is God, the One’ a hundred times when going to bed, God will forgive the sins of fifty years.”