Wire gauge is about amperage – the higher the amps, the larger gauge you need.
Wattage (the amount of power you are using) = Volts * Amps.
The power lines above your house have **much** higher voltage than what your appliances use – 110 kV in the overhead lines vs 110V in your household wires (assuming you are American). The box on the poll is a transformer, which steps the voltage down from transmission levels to what your house can use.
So, back to our equation, the super high voltage means fairly low amperage.
Low amperage means thin wires.
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