Potassium chloride, having such a high acidity, instantly throws off the pH balance in the bloodstream. Since pH is used to goven nueronical impulses that control muscles, it throws the nervous system into an "unrecoverable state of shock". Unrecoverable because once that potassium chloride hits the heart, the cardiac muscles sieze up from the unbalanced pH level. Without the heart pumping, and the entire cardiac nervous system shut down (not to mention all the nerves that the potassium chloride encountered along the way), the brain is deprived of oxygen, as is the entire rest of the body, resulting in tissue necrosis. After several minutes, the necrosis is irreversible, and the inmate is pronounced dead.
This happens so quickly because the potassium chloride is injected directly into the vein. Within two pumps, the majority of that salt will be pumped directly into the heart. The actual necrosis and dying part takes another two-to-three minutes. But, for all intents and purposes, if that gets into your heart, you're dead meat.