Short answer
Wearing sandals was the Prophet's ﷺ habit, and he told his Companions to wear footwear often, so it is a sunnah of habit. But shoes, trainers and boots are equally allowed, and no type of footwear is required. Many Muslims prefer sandals simply because they are easy to slip off for wudu and the mosque.
Wearing sandals like the Prophet ﷺ is a sunnah of habit, rewarded when done to follow him. Any clean, lawful footwear is permitted, and the sunnah manners are to put on the right first, take off the left first and not walk in one shoe.
The four schools agree on this.
Explanation
Jabir heard the Prophet ﷺ say on a military expedition: wear sandals often, for a man is like a rider as long as he wears sandals. Ibn Umar reported that the Prophet wore tanned leather sandals without hair on them, and Ibn Umar wore the same. Sandals were the normal footwear of the Prophet and the Companions.
The ruling. This is a sunnah of habit: following it out of love is rewarded, but leaving it is not blameworthy. Shoes, trainers, boots and leather socks are all fine. The point of the hadith is protecting the feet: footwear lets you walk far without being worn down.
Why many Muslims wear them. Sandals are easy to take off five times a day for wudu and to slip on and off at the mosque door. In hot countries they are also simply practical.
Manners. Put on the right first, take off the left first, and do not walk in one shoe; see the Sunnah way of getting dressed. Praying in footwear is covered in can you pray with shoes on, and wiping over leather socks in wiping over socks.
Avoid pigskin footwear; see can Muslims wear leather.
The evidence
Quran, Surah Al-Ahzaab 33:21
لَّقَدْ كَانَ لَكُمْ فِى رَسُولِ ٱللَّهِ أُسْوَةٌ حَسَنَةٌۭ لِّمَن كَانَ يَرْجُوا۟ ٱللَّهَ وَٱلْيَوْمَ ٱلْءَاخِرَ وَذَكَرَ ٱللَّهَ كَثِيرًۭا ﴿٢١﴾
There has certainly been for you in the Messenger of Allah an excellent pattern for anyone whose hope is in Allah and the Last Day and [who] remembers Allah often.
Following his habits out of love is rewarded.
Sahih Muslim 2096
Sahih... استكثروا من النعال فإن الرجل لا يزال راكبا ما انتعل ...
"Wear sandals often, for a man remains like a rider as long as he wears sandals."
The Prophet encouraged wearing footwear.
Sunan Abi Dawud 4210
Grading to be checked... أن النبي صلى الله عليه وسلم كان يلبس النعال السبتية ...
"The Prophet used to wear sandals of tanned leather."
Describes his own footwear.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Taharah, sunnahs of wudu. The Prophet's dress and footwear belong to the sunnah of habit.
- al-Fatawa al-Hindiyya, Kitab al-Karahiyya, chapter on clothing. On footwear and the manners of putting it on.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. 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.