Example: People in cars often provide hand signals to communicate their dissatisfaction with your driving. They are trying to get your goat, so just smile and wave.
Origin: This may be a mispronunciation of "get your goad". A goad is a pointed rod used to urge on livestock. A modern equivalent of a goad is the cattle prod. To goad is to stimulate into action. The phrase "goad you on" comes to mind. To "get your goat (goad)" then is to be successful in stimulating a response.
Alternatively,
The word gut down through the years was altered to goat. When something gets your gut, it upsets you and ties your stomach in knots.
Example: Revenue has been going South every since we hired the new Sales Manger.
Origin: South is associated with the downward direction, because that is how it is depicted on maps. North is up, south is down. Hence going south is going down. And down is pretty universally associated with poor performance (falling apart) as in airplanes, swimmers, bicycles, and revenue.