When you watch a top player, they seem to have almost supernatural reflexes. The other guy will crush the ball, and whether he gets it back or not, he always seems to react instantly. How does he do this?
The secret is he's not reacting to the incoming ball; he's reacting to the opponent's forward swing. If you wait until the opponent hits the ball before you react, you will have human reflexes. If you instead constantly watch opponents and try to react to where they are going from their forward swing, you will develop supernatural reflexes - or seem to.
You don't have to do this with every shot, only when the opponent is attacking strongly, or when you are trying to cover most of the table with your stronger side (such as the forehand). In both of these cases it's important to react and move quickly, before the opponent actually hits the ball.
Some opponents do last-second changes of direction, so you have to learn at what point the opponent is committed to a shot and direction. Usually if an opponent tries to be deceptive about this he has to slow down his shots, so you don't have to react as quickly anyway, and so can wait longer. But at some point in every player's forward swing he has to commit to a direction, and so it is your job to figure out when that is, and learn to reflexively react to that.