They're called "sleeve garters" and their original purpose was to protect sleeves, keep them out of the way when doing dirty work.

They have a much longer history, but in the period you mention (old west) they surged in popularity due to the appearance of ready-made shirts. Previously, shirts were usually either made at home by housewives, or by the local tailor or seamstress. Mass market production in the 19th century produced cheap factory made shirts that became very popular.

Unfortunately, they didn't come in the huge range of neck/chest/sleeve lengths we have today, so adjustments were needed. In particular sleeves were deliberately made extra long because even people of the same height have a range of arm lengths, and the factory picked the long end of the range so it wouldn't leave anyone's arms bare. But this meant that many men wore shirts with disproportionately long sleeves and needed a means to shorten them.

Sleeve garters were used to draw up the sleeve to proper length and fix it in that spot. Occupations where long sleeves could get in the way of work and get wet or dirty (like bartenders) were more likely to wear them to keep sleeves out of the way.

You also see old west card sharps wearing them, probably also to keep sleeves out of the way and reassure other players that they weren't carrying cards up their sleeve. And you see gunslingers wearing them, which might be a practical method of ensuring that the sleeve didn't get in the way of a fast draw, or maybe just a fashion statement as in "look, I'm a real professional".