The phrase "don't look a gift horse in the mouth" means that one should not criticize or question the value of a gift. The origin of this phrase is rooted in an old practice related to assessing the age and health of horses. 

When buying a horse, it is customary to examine the horse's teeth to determine its age and health. The condition of a horse's teeth can reveal a lot about its overall well-being and how long it might live. By counting the teeth and looking at their wear, one could estimate the horse's age and thus its value. 

Applying this practice to a gift horse (a horse given as a gift) is considered rude or ungrateful. Since the horse is a gift, it's inappropriate to inspect its teeth to assess its value, as it implies doubt about the worth of the gift and ingratitude toward the giver. 

The phrase has been in use for centuries. One of the earliest recorded uses is by Saint Jerome in a letter dated around 400 AD, where he wrote, "Never inspect the teeth of a gift horse." This indicates that the concept was already a known proverb even in ancient times.