Oshibori are damp hand-towels that are given out at restaurants. Commonly one of the things that people miss about Japan when they return home. I love them to. Most cheaper places have plastic wrapped paper oshibori, but many restaurants use cotton ones. These are delivered typically in the morning before most restaurants open for lunch where they are put into heaters or coolers, depending on the season, for customers. Companies like KCC deliver the fresh wrapped oshibori for the previous days used ones for washing and reuse.
This photo was taken in around 10:00 AM in Umeda.