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. 

Jekyll and Hyde, of necessity, die at the same time; because they share the same body, one cannot die without the other dying as well.  

When Mr. Utterson came to Dr. Jekyll's door, he hears Hyde's voice and determines to break it down in order to prevent Hyde from killing Jekyll (not realizing that they are the same person).  When he and Poole chop through the door, they see a laboratory, and "Right in the midst there lay the body of a man sorely contorted and still twitching."  It was Edward Hyde, wearing clothes that clearly belonged to Dr. Jekyll because they were too big for Hyde.  Hyde held a "crushed phial" that had contained some poison, the smell of which still hung in the air.  Hyde had killed himself rather than be arrested and hanged since he could not return to Jekyll's shape without the correct chemicals that he was never again able to procure.  There was no trace of Jekyll because he had been completely taken over by Hyde, and Utterson and Poole assume that Hyde had buried him under the floorboards (until Utterson reads Lanyon's and Jekyll's narratives and learns the truth).