In the beginning of the story, we learn that Roderick was feeling physically and mentally ill and that is why he sent a letter to the narrator, his boyhood friend. Once the narrator arrives at the house, he sees that Roderick is paler than he used to be and that his senses are hightened; and also that his sister Madeline is ill of some mysterious sickness.
Over the course of days, the narrator tried to cheer Roderick writing lyrics to his songs, reading him stories, but nothing seems to work. Over the days following Madeline's death and burial, Roderick seems even more nervous and mentally unstable, until one night he knocks on the narrator's door, completely hysterical. The narrator tries to calm him by reading him another story, but when they hear some noises, Roderick finally loses his mind. He says that Madeline is the one knocking on the door, which is confirmed when the wind blows it open. Madeline attacks Roderick, who dies of fear while the narrator escapes from the House of Usher, which crumbles to the ground.
There are several possible causes for his illness, but I would focus on the mental aspect. Both Usher's seems to be two sides of the same coin: Madeline lack of physical strenght reflects Roderick inability to tell reality from fantasy. He is not afraid of a particular thing, he is afraid of fear itself, and he focalizes it on Madeline. Also, we know that Roderick has become a recluse, never leaving the house. His identity could be so intermingled wih the physical house and with his sister, that the idea of the dynasty dying is what brings the illness. The House, as the dynasty, is deteriorating so when they die, the House crumbles.