At almost every meal/snack/drink in any restaurant/cafe/bar, a plastic-wrapped wet towelette will be given to everyone at your table.
Most common are the thin wet-tissue variety. Occasionally, you may get a sealed wet cloth towelette (sometimes even microwave heated!).
They are easy to rip open though sometimes it’s difficult to unfold the very thin ones.
And they are for wiping your hands, NOT YOUR FACE!
Use them before you touch any food or plates or chopsticks or cutlery. Then refold them neatly and leave them to one side for the duration of your meal.
Japan Tourist Tips 