The phrase 'to quarterback' or 'quarterbacking' means to take control of something. This is a reference to the quarterback position in football, often considered the most important position in the game.
A quarterback must know all of the in-and-outs of both football teams playing and the rules of the game. They are considered an 'on-field coach' as they must run the offensive unit in order to win games. The phrase 'quarterbacking' or 'quarterback' something means to take charge of something or a situation much like a quarterback in the game of football.
The quarterback is the player who tells the offense what plays to run and handle the ball the most out of any player on the offense. The ball is almost always snapped to the quarterback who either throws the ball, runs the ball, or hands the ball to a running back to run the ball.
Due to the nature of the position, the quarterback is forced to take on a number of responsibilities both within the offense and the team as a whole. Being able to handle the large amount of responsibilities is very difficult and only a select few are capable.
The quarterback is thought to be the most important position in football due to how much is asked from them. A quarterback is the 'coach' or leader of the offense as well as the entire team, other than the head coach.
They need to be able to handle the football on every down, even when it is not a passing play. On top of that, they must call the play, read the defense, and control the tempo of the game, all before the play even starts!
The importance of a quarterback is highlighted by the difference in success between teams with a great quarterback and teams with an average quarterback. A major example of this is the New England Patriots, whose past two decades have been filled with 5 Super Bowl Championships, thanks to legendary quarterback Tom Brady.