عن أبي هريرة رضي الله عنه أن رسول الله صلى الله عليه وسلم قال: لا يمش أحدكم في نعل واحدة لينعلهما جميعا أو ليخلعهما جميعا وفي رواية أو ليحفهما جميعا متفق عليه
Abu Hurairah (May Allah be pleased with him) said: The Messenger of Allah (PBUH) said, "None of you should walk wearing one shoe; either wear them both or take both off .''
[Al-Bukhari and Muslim].
وعنه قال سمعت رسول الله صلى الله عليه وسلم يقول إذا انقطع شسع نعل أحدكم فلا يمش في الأخرى حتى يصلحها رواه مسلم
Abu Hurairah (May Allah be pleased with him) said: I heard the Messenger of Allah (PBUH) said, "When the lace of one of the shoes of one of you snaps, he should not walk with the other until he has got the lace repaired.''
When one reads the commentaries of the scholars regarding these hadiths, they came up with a variety of explanation for its wisdoms.
I quote Dr. Abu Ameenah Bilal Philips:
"Thus, there may very well be some medically related reasons for the Islamic prohibition on walking in a single sandal or shoe - which is yet to be discovered. Perhaps, it may be due to a temperature imbalance created by the temperature difference between the covered foot and the bare foot when one walks on ground colder or warmer than his body temperature."
[Tafseer Soorah al-Hujuraat pg. 182]
Perhaps he has a valid point. However, it is immediately apparent to those who have some knowledge regarding ergonomics, that this prohibition has a lot to do with ergonomics of walking or running with only a shoe and a bare foot and that this will lead to health problems in the future due to continued imbalance of weight distribution between the two feet.
"Proper ergonomic design is necessary to prevent repetitive strain injuries, which can develop over time and can lead to long-term disability."