Put simply a fatter or better quality wire will allow more electrons down it. That is more amps, simple as that. When a lot of power is asked for, as headlights do, the wide wire will allow them to flow through maintaining full pressure (voltage), rather than causing a bit of "congestion" (resistance) meaning the amps you want don't get through at the desired voltage.
For reference:
Volts X Amps = Watts.
Amps = Quantity of electricity
Volts = Pressure of that electricity
Watts = Total amount of "work" it will do per second.
A Watt is one amp at one volt, consumed over 1 second. Or... Enough "power" to lift a 100 grams of weight 1 metre high in 1 second.
Oh and a Joule is a watt without the time constraints.

Ali