From the Smithsonian Magazine:

Heavy metals including gold and silver are antibacterial, but copper’s specific atomic makeup gives it extra killing power, Keevil says. Copper has a free electron in its outer orbital shell of electrons that easily takes part in oxidation-reduction reactions (which also makes the metal a good conductor). As a result, Schmidt says, it becomes a “molecular oxygen grenade.” Silver and gold don’t have the free electron, so they are less reactive.
Copper kills in other ways as well, according to Keevil, who has published papers on the effect. When a microbe lands on copper, ions blast the pathogen like an onslaught of missiles, preventing cell respiration and punching holes in the cell membrane or viral coating and creating free radicals that accelerate the kill, especially on dry surfaces. Most importantly, the ions seek and destroy the DNA and RNA inside a bacteria or virus, preventing the mutations that create drug-resistant superbugs. “The properties never wear off, even if it tarnishes,” Schmidt says.