The advantage of transmitting electricity at high voltage is that is minimizes the power loss due to resistance in conductors. Power is dissipated as heat. The relationship between power, current and resistance is P=I^2R. The power loss in the transmission line is be proportional to I^2. The relationship between Power, current and voltage is P = IV. Voltage is inversely proportional to current. Increasing the voltage decreases the current, which decreases the power loss.
Resistance is friction caused by the motion of electrons in the conductor. One challenge of transmitting electrical energy over long distances is that resistance increases with the length of the line. Increasing the cross-sectional area of the line to reduce resistance would increase the weight of the line, making it harder to support.
High voltage is decreased for local lines by using step-down transformers. Voltage and current are reduced by the same amount so power, which is IV, isn't changed.