I use it because it puts a protective coat on metal surfaces. Your hands have oil and a mild acid on them as well as moisture. The cloth removes fingerprints and keeps the windscreen clean and prevents rust and corrosion from forming.
Many wind screens and microphone bodies are nickle plated or stainless steel. Regardless of what you may have heard, both of these surfaces will rust, albiet slower than uncoated carbon steel. Over time, even the moisture in the air can produce very fine surface surface rust that is difficult to see, but nevertheless is present.
Also great for blued-steel firearms.