Run support per nine innings measures how many runs an offense scores for a certain pitcher while that pitcher is in the game. That number is then set over a nine-inning timeframe. So the stat essentially answers the question, "How many runs of support does a pitcher receive per nine innings?"
In no way is RS/9 something a pitcher can control. (On the mound, at least. In National League parks, a pitcher can help his cause as a hitter.) Instead, RS/9 is way of adding context to a pitcher's win-loss record. Does a given pitcher's winning percentage seem a bit too high or a bit too low given his other stats? RS/9 is often the culprit.
It's important to note that for this metric, run support constitutes only the runs that are scored for a pitcher while he is in the game. A few other run-support metrics will take into account how many runs a team scores for its starting pitcher over the course of an entire game. In that vein, RS/9 also works for relief pitchers (although those numbers should be taken with a grain of salt, because relievers have such small sample sizes in terms of innings pitched).
In A Call
"average run support," "support per nine," "runs of support per nine"