That is, it will eventually change its direction of travel, but just because it is accelerating in the opposite direction of the current vector does not mean that it has changed direction; yet.
I think this is the core of your confusion. The object is not accelerating in the opposite direction of the current velocity. Try drawing a diagram of the two vectors and you'll see what I'm talking about.
When the player hits the ball with top spin, it makes the ball, well, spin.
By spinning, the ball will modify the airflow around itself and thus create an air pressure profile which will deflect the ball : this is the Magnus effect.
So by applying top spin on the ball the way tennis players do, the ball is rotating in the direction of the trajectory. This will bend the trajectory downwards. If you look at the ball's speed as a vector, the vertical component of the top-spun ball's velocity is greater than the normal served ball.
You simply direct the ball more vertically into the ground with a top spin. So after contact with the ground, the ball with top spin will leave the ground more vertically than a normal ball.
Since you are familiar with the sport, you might also have noticed that after contact, the top-spun ball will slightly accelerate towards you. This comes from the fact that part of its rotation has transfered itself into horizontal momentum. If you let a spinning ball fall vertically on the ground, after contact, it will fly out flat in some direction. I think this is also one of the reason why top spin serves are so hard to handle.
Best Answer
One way to explain it that makes sense to me is that the backwards spin on the ball is a force pushing the ball toward you. The one time applied force of your finger that creates the backward spin also pushes the ball forward, but the backspin stays almost constant. At the moment you spin the ball, the forward force is greater then that of the backspin, so the ball moves away. After time zero, the backspin is the only force acting horizontally (you aren't still touching the ball, so there is no applied force) and so the ball slows, then returns because of a negative acceleration.
If you were to graph this motion as Position (away from you) vs Time, you would get a graph that looks similar to the motion of a ball thrown upwards and pulled back by gravity.