A gamekeeper is someone who is hired to breed and protect game. A poacher is someone who hunts for game illegally. A poached egg is a type of egg that is cooked more delicately, in simmering water instead of boiling water. This joke is playing on the word "poacher" because the word "poached" can relate to eggs, and not just having hunted something. 
Getting this job managing a country estate has put me off fried eggs . I ' m a gamekeeper turned poacher .