Why do I have three buttons on the sleeves of my sports jacket but don't have three buttonholes or any possible reason to want three buttons on my jacket sleeves? Let's see you answer that one.
-- Not Einstein, San Diego
Like most questions for which no one has a real answer, there are two answers, the rational-sounding one and the goony one. Life being what it is, the goony one is probably correct. No matter which you choose, the buttons are basically vestigial decorations, a sort of fashion appendix. If you like your explanations tidy and tasteful, then the buttons are left over from the days when wealthy men buttoned tight-fitting coat sleeves over their lace-cuffed shirts. Buttons actually were a fashion decoration mania a few centuries ago. The ruder, cruder explanation credits either (a) the French or (b) the Prussians with covering the cuffs of military jackets with big buttons to keep the troops from wiping their noses and mouths on their sleeves.