@fBoyle-- This idiomatic phrase can be used in different ways. I usually use this phrase when someone suddenly becomes very friendly with me. For example, I have a coworker who doesn't like me and who is usually distant from me. When she suddenly becomes very friendly and starts speaking to me, I know that she has an axe to grind. So I know that she needs my help to do something.
So you can use this idiom to explain why someone is doing something that they are unlikely to. If an individual is resentful of someone or something for example, they may make excuses to bash them. To the outside world, it can be difficult to understand why this person is acting this way. But if you know that they have a past grievance, you can conclude that the individual has an axe to grind -- an issue to settle.