To refuse something is, at the deepest level, to pour it back. The word comes from Old French refuser, from Vulgar Latin *refūsāre, rooted in Latin refundere — 'to pour back', from re- ('back') and fundere ('to pour').
The pouring metaphor is invisible now, but it shaped the word's logic. An offer is extended like liquid in a cup. To refuse is to tip it back — to return it undrunk.
The noun refuse — meaning rubbish or waste — shares the same origin but through a different grammatical route. Old French refusé was the past participle: 'refused things', 'rejected matter'. Refuse is what remains after everything wanted has been taken. The word carries judgement: rubbish is not merely unwanted, it is actively rejected.
Latin fundere is among the most productive roots in English. From it pour confuse ('pour together' until muddled), diffuse ('pour apart'), infuse ('pour in'), profuse ('pour forth'), transfuse ('pour across'), and refund ('pour back' — specifically money). The common thread is liquid movement: mixing, spreading, filling, returning.
The connection between refuse and refund is direct. Both mean 'to pour back'. One pours back an offer; the other pours back payment.