To feed a grievance would be to do something that would make someone even more upset about something, when they were already upset. In other words, they had a grievance - the circumstances around him getting a dog made the grievance stronger. Pet-rescue activists are angry because he had promised that he would get a dog from a shelter, but he didn't.
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KateJS I've no idea who Rush Limbaugh is, sorry!Better watch your back. Somebody once said he was irrelevant, and evoked his eternal rage.