This is a widespread myth and has some variations, sometimes the spoon must be silver other times the material doesn’t matter. It’s unlikely that the spoon makes any difference. There is some suggestion that the metal will absorb more heat energy and keep the air in the bottle colder, which may aid the preservation of the sparkle in your wine.
The temperature is certainly important though. Colder liquids can hold more gas, in the case of sparkling wines, carbon dioxide (CO2). This is because the higher the temperature the more energy the atoms or molecules in the liquid and gas have, making it more likely that the gas molecules can escape the liquid. So the lower the temperature the lower the energy and less CO2 can escape. So putting the wine in the fridge is the most beneficial thing. The best thing to do, though, is finish the bottle!