The simple answer is dehydration kills you by causing the blood flow to become completely ineffective.
Dehydration limits the amount of water in the bloodstream causing an increase in blood viscosity, and thereby an increase in blood pressure. As this process gets more extreme the body has a difficult time getting the blood to the places it needs to be, and will limit blood flow to non-essential organs (kind of similar to what happens when it's cold outside and your fingers go numb). At the same time waste is beginning to build up as the kidneys cannot filter the blood effectively, and the internal temperature rises because of the lack of sweating. This process gets more extreme until one of your essential organs cannot take it anymore and shuts down, likely the liver first.
Most everything I got for this answer was from a PopSci Article. Although there is also a nice 1935 study with many measurements from real people.