How does the gain factor adjustment on this current sense board affect the output voltage range?
With a gain of 1 you can measure 0 to 5 amps as 0 to 5 volts.

With a gain of 10 you can measure 0 to 500mA as 0 to 5 volts.

With a gain of 20, it's 0 to 250mA

etc.

The chip doesn't have a resolution - it's an analogue device.  It's purely down to the receiving device what the resolution is.  With the addition of the op-amp the upper end of the range can be modified.  So, say you have it set to a gain of 20, to measure up to 250mA, and then have it connected to an Arduino, with 10-bit ADC.  The resolution would be 250/1024 = 0.244mA steps.