This is a phrase in football which refers to a running play which is initiated from the shotgun formation. There are a variety of formations the offense may assume prior to starting a play. One of the more popular ones in the modern game of football is the shotgun. In this formation, the offensive line lines up normally, but the quarterback is positioned three to four steps behind the center. When the ball is hiked, it travels a short distance, which is where the name "shotgun" came from. This formation was originally designed to allow the quarterback more time and space to pass the ball. However, it has adapted over time and now many teams will run the ball from this formation.
Running the ball out of the shotgun formation is a fairly recent development in the game of football. Since the quarterback is far back from the line of scrimmage, when he hands the ball off, the rusher will also be far back. This was formerly seen as a disadvantage, but once athletes got faster and teams figured out how to execute it successfully, it became a lot more popular. A benefit of running from the shotgun formation is it keeps the defense on their toes, as the formation will not give away whether the ball will be passed or handed off.