James,
It doesn't work that way with ampere.
Volts is the electrical potential, and you have to use a PSU (Power Supply Unit) that outputs the desired voltage, but the amps rating on a PSU is a max value of how many amps the PSU can supply, not how many the board will get.
The load (in this case the Galileo board) decides on how many amps it will pull out of the PSU. In base configuration without any shields or Mini PCIe cards installed, it's about 550mA that the board uses at 5V. If you have it do a lot, it will likely go up, and the same is true if you add shields that uses a lot of power.
The BEC circuit is just a PSU that uses a battery as power source instead of plugging into a wall outlet.
Hope it helps!
/Thomas