What are the advantages of increasing the voltage in a circuit?

For appliances that need a lot of power, the advantage of a higher voltage is that less current is needed, because power = voltage x current. That means that the cable doesn't need to be so thick.

It also means that for a given percentage of voltage drop in the cable, the allowable volt-drop in the cable is higher when the supply voltage is higher. This also allows the use of a thinner cable especially when the cable is 100 ft long or more.