To remove what is not wanted until only the wanted remains. Refining subtracts rather than adds.
Re- (Latin, again) attached to fine, from Old French fin (an end, a limit), from Latin finis (end, boundary); to refine was first to bring metal to its final, purified state, subtracting until only the finished remained.
To remove what is not wanted until only the wanted remains. The root's ending is the point: refining is a subtraction that stops at a finished state, not an addition that keeps going.