I got that phrase from my sister.* In a family, it means to stir things up, generally to foment or create attention, often done by people who are completely unaware of what they are doing. e.g. "You may have the smallest family, but your children are the best" -- said to one family member while others are around and with implied criticism of the person with the small family.
Or consider: "Trisha, Jaydeen was complaining about the way you didn't have vegetables with your meals" said to Trisha when Jaydeen wasn't complaining, but had only mentioned that Trisha had a bar-b-cue.
Often, too often, the person who "stirs the pot" isn't trying to cause harm, but is only focusing attention on themselves, coloring everything that they say or that passes through them to generate or tell a better (i.e. more interesting) story.
The problem is that there is very real fall-out from that behavior in terms of hurt feelings, trouble and conflict, fall-out that far out-weighs the joy anyone has in telling a more interesting story or having a more riveting dialogue.
The solution to someone who stirs the pot is to learn to talk around the pot stirrer, often a person who is otherwise wonderful, but who had very dysfunctional communication patterns and who often attempts to centralize communication patterns to go through them. In fact, if there is someone in a family who steers communication so that it goes through them, you probably have someone who will stir the pot.
Compare pot stirrers to drama queens who draw attention to themselves, rather than use messages to draw attention to what they are saying. Many only stir the pot within a family or when communicating with only one sex (often only daughters and daughter-in-laws or only sons and son-in-laws). Whenever there is conflict created because of things people are hearing second hand, they need to stop and ask if someone is stirring the pot in the middle of the communication stream rather than someone on the other end causing a problem.
Once you've got the personality type identified, you will spot them at work as well as within families. They are the people who gossip to generate emotional response rather than to pass along information and the solution is the same. Find ways to talk around them.
(* I was asked if all my best ideas come from either my wife or my sister. No. I get some of them from my brothers. The term is used in other contexts as well, see: Stir the Pot as applied to political and community discourse. In a family it means keeping things stirred up, in politics it means bringing things back to the surface).