Pump action hand washes and white soap bars contain an emulsion, when we dry our hands it comes into contact with the towel. When the towel goes in your washer this emulsion attracts the dirt from other garments. Then at the end of the cycle, clothes have dark patches left on them. These may look worst when you hold the garments up to the light.
Make sure that all soap is rinsed of the hands thoroughly before using a towel. Unfortunately this type of damage can't be reversed.