There are two meanings to the phrase “an axe to grind.” The first meaning is the traditional American one, which means having an ulterior motive or personal reasons, other than the obvious, for doing something. The British meaning is to hold a grudge or a grievance against someone or something.
The term comes from the grinding of axes using a grindstone. Axes were first made of wood and stone, then came to be composed of the best metal available to the person. They are used for splitting wood, felling trees and various other things. They are also a weapon of aggression. Axes are sharpened using a round grindstone that is rotated on an axle using a foot pump.
Origins of the term in both the US and Britain are commonly accepted to have come from America. The British sense of the idiom seems to have been an addition or variation to the American idea. There is little evidence of the notion outside of America before the 19th century.
The origin of having “an axe to grind” comes from one of two men in Philadelphia. Charles Miner wrote a cautionary tale about his childhood. In the tale, printed in 1810, he was duped into grinding an axe for a man using a grindstone. Once Miner finished the task, the man left without even saying "Thank you" or rewarding Miner for his hard work.
Naturally, Miner held a grudge of sorts and used the metaphor to warn others of ulterior motives and self-interest. His tale led to him saying in 1812, "When I see a merchant over polite to his customers…Thinks I, that man has an axe to grind.” The hidden motive for the merchant is profit and the metaphor can be taken to mean someone who is nice in order to get what he wants.
The other man who may have coined “an axe to grind” is Benjamin Franklin. In his posthumous autobiography, Franklin gives two similar cautionary tales to Miner. His autobiography was published 20 years earlier than Miner’s. While many believe that Franklin invented the term “an axe to grind,” he never used the term itself.
Both stories may be apocryphal. This means they are drawing on an older tale, maybe from England, of someone asking someone else to grind an axe, but having an ulterior motive for doing so. The difference is that Miner invented a catchy idiom or phrase to capture an older idea. This might explain why when the term was introduced to Britain, it came to mean a grudge or grievance, which might have been the original meaning of the cautionary tale.