It is clear from the story that what drives Paul to making himself ill is his repeated riding sessions on his rocking-horse. Paul is driven to finding out the results of the horse racing by his mother's greed for money, echoed in the whispers in the house, "There must be more money!" One could argue that Paul does this in an attempt to gain the affection of his mother as she is presented as distant, aloof and someone who finds it impossible to love her children.
When the chance comes to gain some serious money because of the Derby, a very big horse race in England, Paul is determined to find out the winner, even though he is obviously getting weaker. Note how he is described:
The Derby was drawing near, and the boy grew more and more tense. He hardly heard what was spoken to him, he was very frail, and his eyes were really uncanny.
You might also want to trace the way that Paul's eyes are described through the story, and how the more he uses the rocking horse to predict the winner, the brighter they blaze. Note the last time he uses it, his eyes "blazed", before he crashes to the ground.
Paul then dies because he gives himself over more and more to the rocking horse and finding out the winners of races so that he can gain money for his mother.