A silver polishing cloth is a soft cloth with special chemicals to remove and prevent tarnish. Often they contain a solution of mercapto ester and amine salt to clean, polish and protect silver.
These cleaning solutions help to break down the sulphides that cause silver to turn black and dull. Some of these sulphides come from the air but also from perfume, soaps and lotions as well as some foods. Even plated silver tends to react with the sulphides so unplated items such as jewellery and cutlery will often require more attention to keep them brighter.
An alternative to a silver polishing cloth is an ultra sonic cleaner which bombards the liquid with ultrasonic sound waves that create millions of minute bubbles that scrub the surface. However they are not really practical for individual items as the process is time consuming and they can leave microscopic scratches on some delicate surfaces.
For heavily tarnished silverware or jewellery you can try this recipe. Line a baking pan with aluminium foil (use ceramic or glass bakeware never metal) and fill with warm water then add baking soda. Soak the silverware or jewellery in this for a minute then take them out with kitchen tongs, rinse under lukewarm water and dry them on a paper towel. This is especially effective with silver chains – just be careful not to over tighten the links and don’t touch any of the other jewellery as it will mark the plated finish.