'refine' originates from Middle French/Old French (e.g. 'raffiner' / 'refiner'), ultimately from Medieval Latin/late Latin formations where the prefix 're-' meant 'again' and the stem related to 'finis'/'finire' gave the sense of 'make fine' or 'bring to an end'.

'refine' changed from Old French 'raffiner'/'refiner' into Middle English (e.g. 'refinen') and eventually became the modern English word 'refine'.

Initially it meant 'to make fine or purify' (especially in context of metals or foodstuffs); over time it broadened to mean 'to improve or make more precise' in many abstract contexts.