What are the advantages of transmitting electric power at high voltage levels?

For a given amount of power, the higher the voltage the lower the current. This is necessary to minimise the voltage drop along the transmission line, and to reduce energy losses which are proportional to current squared.

The advantage of transmiting electrical power at high(er) voltages is that you can use smaller conductors, since the current flow would be less for the same amount of watts. Watts = Volts times Amperes