There's a good reason why a loosely strung tennis racquet makes the ball travel fastest: the strings are livelier than the ball itself. By "livelier" I mean that the strings store and return energy more efficiently than the ball does. To maximize the ball's final speed, you want the strings to do most of the bouncing, not the ball. To understand this effect, let's examine the bounce that occurs when the ball and racquet collide.
As the two objects hit, they begin to dent. Kinetic energy (energy of motion) in the ball and racquet becomes elastic energy in the dented surface of the ball and the deformed surface of the strings. Eventually, the ball and racquet reach the same velocity and dent no further. They then begin to rebound. The ball snaps back to its spherical shape and the strings return to flatness. As they undent, they convert their elastic energies back into kinetic energy and push apart. The ball flies over the net and hopefully past your opponent.
Where liveliness enters this picture is in the efficiency with which the ball and racquet store and return kinetic energy. In a lively object, most of the original kinetic energy present before the denting process begins returns as kinetic energy after the denting is over. But in a dead object, much of this kinetic energy is lost as thermal energy—the object becomes warmer. A beanbag is the ultimate dead object—it doesn't bounce at all. It turns out that a tennis ball is fairly lively, but not as lively as the racquet strings. These strings may not be spherical like a ball, but they can bounce and do it very well. Nearly all of the kinetic energy stored in the strings during a bounce returns as kinetic energy when the bounce is over.
Because the strings are livelier than the ball, you get the most "power" out of the racquet when you string it loosely. That way, as the ball and racquet collide, the strings dent deeply and do most of the bouncing. The fraction of the original kinetic energy stored in each object is proportional to how far each object dents. With the strings denting much more than the ball, the strings store most of the energy and are responsible for most of the bouncing process. They handle this energy efficiently and the ball comes off the racquet at top speed. A more tightly strung racquet makes the ball do most of the bouncing. The ball does this adequately but it doesn't leave the racquet quite as fast as it would if the strings were looser.
However, there is an advantage to tighter strings. Because a tightly-strung racquet is relatively flat throughout the bounce, the direction the ball takes as it rebounds is easy to predict. When the strings are looser, the racquet surface deforms significantly during the collision and the ball's final direction depends on exactly where it hit the racket head. As noted on your racquet's instructions: a tightly strung racquet gives you more control over where the ball goes, but a loosely strung racquet makes the ball travel faster.
Answered by Lou A. Bloomfield of the University of Virginia.