Mr. Hyde is the dark side of Dr. Jekyll; he is part of the same man, and the two share a body. In death, then, only one body is found.
The darker side of Dr. Jekyll eventually takes over and the man completely becomes Mr. Hyde. This is why in the end the dead body of Hyde is discovered wearing the clothes of Dr. Jekyll.
Earlier in the novel, Dr. Jekyll develops a chemical mixture that allows him to separate his moral nature from his immoral one because he has wondered if he could indulge his dark side without the pangs of conscience. However, what Dr. Jekyll has not figured is the fact that with each action, the darker side develops, and soon the evil grows inside Mr. Hyde until he commits a murder. Then, when he attempts to return to his better self, he finds that he cannot. Desperate for more of his potion, in the persona of Mr. Hyde, he has Dr. Lanyon procure the drugs from his laboratory. When Hyde takes the potion and begins to transform right before Dr. Lanyon, Lanyon is horribly shocked. He dies shortly thereafter.
The horror, then, in this tale of the secret of Jekyll and Hyde is that both parts of this man are trapped; they each fight for dominance over the other.
This brief condescension to my evil finally destroyed the balance of my soul...
writes Jekyll in a letter. This is why Hyde/Jekyll dies when the good side tries to rid itself of the evil side.