A tennis player is allowed to serve two times. If your first attempt (serve) results in a fault, the player is allowed a second attempt. When you commit a fault, in many, if not all competitions, you are discouraged to go and retrieve the ball that resulted in your fault. This would cluster the job of the the ball-boys on the court who's job is to collect dead balls.
Proper etiquette and the rule that prohibits wasting time dictate that a player serving a first serve should have a ball in his/her pocket so that if a second serve is required, the player serving doesn't need to go find another one. It isn't a big deal for most tennis matches, but there are prescribed time limits between points.
Plus, if you watch closely, you'll see that players filter through balls. The ball runners provide three balls, the player puts one in his pocket, gives one back to the ball runner, and serves with one.
