“Don’t look a gift horse in the mouth” means don’t question a the value of something that you received for free. In this sense it is like “beggars can’t be choosers“.
A horse’s approximate age can be determined by examining its teeth.
In ancient times horses were often gifts and it would have been extremely insulting for the recipient to examine the horse’s teeth in front of the giver. It would be saying “hey, are you giving me some old worthless horse”?
In modern times, it would be rude to rush to look for a price tag on a gift in front of the giver or ask the price. It is simply not polite.
Example: “If you can earn some extra income in this economy, don’t look a gift horse in the mouth. Take it.”
“Vic was unhappy because he received an expensive sweater from his grandmother for his birthday, rather than a video game. ‘Vic, don’t look a gift horse in the mouth. Remember, beggars can’t be choosers,’ his mother said”.