Why do diabetics have the fruity smell of ketones on their breath?

they're in a state known as diabetic keto-acidosis (DKA), a true life-threatening emergency.

DKA occurs when a diabetic patient becomes hyperglycemic, i.e. they have too much glucose in their blood, but not enough insulin to move it to the liver or cells. This is because those with Type-1 diabetes are unable to produce insulin, and so depend on injections to regulate their blood sugar. Without it, glucose accumulates in the blood.

Initially, the body attempts to excrete the excess glucose via the urine, but this requires water. Eventually, the patient will become severely dehydrated, made worse by the drastic loss of other ions such as sodium and potassium in the urine. At this point, the glucose levels continue to climb as the liver starts to convert more and more of its glucagon stores into glucose, a process that is usually inhibited by insulin. It also starts to release fatty acids.

The fatty acids are broken down into mildly acidic ketone bodies via beta-oxidation, which then causes the pH of the blood to decrease; hence the state of keto-acidosis. The body tries to buffer the acidosis by hyperventilation, 'blowing-off' excess #CO_2#, and it's at this point that the enormous levels of ketones in the patient's blood can be detected in the patient's breath.