This is a phrase used in sports, such as football, to describe the action of a winning team continuing to score, even though they will clearly win the game. When a team is winning by a significant margin, the tradition of good sportsmanship says that a team should stop scoring and simply attempt to run down the clock or keep the ball away from the opposition. In professional sports, such as the NFL, running up the score is not considered as bad as when it is done at the youth level, for example. Since athletes are paid to win, scoring and beating the team is their job and they are usually expected to keep going until the end of the game. However, even though it is more accepted, it is still considered bad sportsmanship if a team runs up the score.
The phrase running up the score is especially applicable to football because of the relation between the game clock, play clock, and the ability of a team to run down both. If a team is ahead and is in possession of the ball, they will usually not try to score if they have the ability to run down the clock. This is accomplished by a special play in which the quarterback kneels down immediately after hiking the ball. The result is an immediate dead ball while the game clock keeps running, exactly as if a running play had ended in a tackle.