@Nancy88,
"Chipping away at" means to do something bit by bit, as in chipping at a stone with a chisel when you want to make a carving. You can chip away at your homework by reading bits of it at a time instead of all at once, chip away at your housework, etc. But...
I don't really get how a person 'chips away' at being part of a company for 24 years. They're either a part of it or they're not; it's just not something you chip away at.
Strange usage of the phrase, in my opinion. Unless the writer's referring to him chipping away at being a faithful, long-time employee? And that would still be strange.