Short answer
Around 25 AH (about 645 CE), Muslims in different lands were disputing over ways of reciting. To keep the ummah united on one text, Uthman had official copies made from the collection kept by Hafsa, sent them to the main cities, and ordered other personal copies burned. The Companions agreed with him.
The Sunni account, from Anas in Sahih al-Bukhari, is that Uthman acted on Hudhayfa's warning, used the Abu Bakr collection, had Zayd ibn Thabit and three others write the copies, and burned other copies so the Quran would not be lost or disputed. Ali praised what he did.
The four schools agree on this.
Explanation
Hudhayfa ibn al-Yaman fought in the campaigns in Armenia and Azerbaijan alongside soldiers from Syria and Iraq. He was alarmed to hear them arguing over how to recite the Quran, each sure his way was right. He came to Uthman and said: catch this ummah before they differ over the Book as the Jews and Christians differed.
Uthman asked Hafsa for the sheets gathered under Abu Bakr after Yamama. He appointed Zayd ibn Thabit, Abdullah ibn al-Zubayr, Sa'id ibn al-As and Abd al-Rahman ibn al-Harith to copy them into several volumes, writing in the dialect of Quraysh where they differed, since the Quran came down in it. He sent a copy to each major city and ordered that other private sheets and copies be burned.
Why burn them? Personal copies often mixed in a Companion's own notes, explanations or readings that had later been dropped. Leaving them would keep the arguments alive. Burning was a way to remove written Quran that would no longer be used, so it would not be thrown away or misused. Today the Hanafi scholars prefer to bury a worn mushaf, while others also allow burning it respectfully: see How do you dispose of an old Quran?
The Companions supported Uthman. Ali is reported to have said: say nothing but good about Uthman, for by Allah he did what he did with the copies only with our counsel. Every Quran in the world today goes back to these Uthmani copies. Read History of the Quran and Uthman ibn Affan.
The evidence
Quran, Surah Al-Hijr 15:9
إِنَّا نَحْنُ نَزَّلْنَا ٱلذِّكْرَ وَإِنَّا لَهُۥ لَحَٰفِظُونَ ﴿٩﴾
Indeed, it is We who sent down the Qur'an and indeed, We will be its guardian.
Allah promised to preserve the Quran.
Sahih al-Bukhari 4987
Sahih... أدرك هذه الأمة قبل أن يختلفوا ...
"Catch this ummah before they differ."
Hudhayfa's warning to Uthman.
Sahih al-Bukhari 4987
Sahih... وأمر بما سواه من القرآن في كل صحيفة أو مصحف أن يحرق ...
"He ordered that every other sheet or copy of the Quran be burned."
Uthman's order for the other copies.
Where the ruling is recorded
- Fath al-Bari (Ibn Hajar), Commentary on Kitab Fada'il al-Quran, chapter on the collection of the Quran. Discusses the date, the committee, the burning and the report of Ali's support.
- al-Bidaya wa al-Nihaya (Ibn Kathir), Events of the year 25 AH, the copying of the mushaf. Places the Uthmani copying in his caliphate.
- al-Itqan fi Ulum al-Quran (al-Suyuti), Chapter on the collection and ordering of the Quran. The classical summary of the two collections.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.