Neutral width is used to keep your ESC at neutral when the trigger on your radio is at the neutral position. It can be adjusted to allow for radios that may not be as precise in returning to the same neutral position when the trigger is released. You're setting a buffer zone, if you will, that says "if my radio is transmitting anywhere within this region, consider it neutral." If you have a radio that does not return to a very precise point every time when you let go of the trigger, and you have a narrow neutral width on the ESC, it can result in the ESC receiving a drive signal when the trigger is at neutral. Obviously, we don't want that.

I think some ESCs call this deadband also.

I don't think it's really considered a tuning tool, it's more a setup function.