How does his reaction reveal what he believes are his "dreams"?

You're referring to Walter? When Walter's mother says that he can have a portion of the insurance money to invest in his own business venture, he is excited and looking forward. Travis enters the room shortly after, and Walter, who is thrilled by his new-found responsibility, tells his son about his hopes and dreams of working in a office, driving a nice car, having a nice house, and sending his son to college. These are the dreams he believes in.