In this story, little Paul is hopelessly striving to, through helping his mother get money, win her affection and love, and to help his entire household to be happy in order to quiet the voices demanding more money. He takes it upon himself to resolve the situation, and to bring happiness through love and peace--he thinks that can be gained through money. His mother led him to believe this in the unfortunate conversation she had where she equated being lucky to having money, and how sadly, their family had no luck. The whispering of the house also drives him further into his unusual quest on the rocking horse.
When Paul says he "must get there," there is probably referring to some metaphorical or symbolic place where his mother his happy and loves him, and where their family has enough money to be happy. "There" could be the fairy-tale ending that all children would hope for in their homes: happy parents who adore their children, peace and contentment everywhere they turn, and a life of play and joy. "There" could also refer to whatever place he goes to in his mind to find the winners of the horse races; we don't know how he gets the names, but he does, somehow. He usually has to ride and ride until he gets the name; so perhaps he is riding towards some destination where the name is given to him. Unfortunately, "getting there" is, in the end, what kills him. All of it was done in the fruitless attept to win happiness and affection through money and luck.
I hope that those thoughts helped; good luck!