Tennis players produce topspin by brushing their strings up the back of the ball. This makes the surface of the top of the ball spin forward (in the same direction as the ball's flight), thus increasing its friction with the air and causing greater air pressure on the top of the ball than on the bottom of the ball. This effect, called the Magnus force, pushes the ball downward as it flies.
Topspin makes the ball drop faster; therefore, it allows you to hit higher over the net at any given forward ball speed. Just by reducing the likelihood that your ball will hit the net, topspin provides a huge advantage, but higher clearance over the net brings a second advantage as well: a ball that descends from a greater height will bounce higher, often so high that your opponent can't generate a powerful response, especially on the backhand side.
If your opponent wants to avoid meeting the ball uncomfortably high, she can either move forward to hit it on the rise or move back so far that she meets it after it has dropped from its post-bounce peak, but the ease of making this adjustment depends, in large part, on how steeply the ball bounces. Bouncing higher doesn't necessarily mean bouncing upward at a steeper angle, because the angle of a bounce is a combination of its upward and its forward components. While a topspin shot that flies higher will also bounce higher, and the steepness of its bounce is increased by the steepness of its descent (explained below), these factors are offset somewhat by two effects of topspin that tend to decrease the steepness of the ball's bounce by increasing its forward speed: the ball's forward spin causes it to slow down less when it hits the court, and topspin allows greater forward air speed than would be possible to keep a flat or backspin ball shot in bounds at the same height.
Topspin shots bounce most steeply on clay, because the ball digs a shallow hole as it hits the court and is thus both slowed down and pushed upward by the front slope of that hole as it exits. This has a huge effect on the clay-court game. Balls are harder to hit on the rise when they are rising more steeply, because it's harder to time the racquet's forward motion to intercept a ball that's spending less time within a given height window, and the uneven clay surface imposes the added difficulty of a somewhat unpredictable bounce. If you play farther back, though, you have to run farther, because you don't cut off your opponent's angle shots as early. To run farther, you need more time, and clay helps you by slowing the ball down. The increased difficulty of meeting the ball early and the relative ease of meeting it farther back explain, in large part, why most clay-court specialists play farther back than those who prefer hard courts or grass.