Japan excels at taking ideas from different cultures and tweaking them to fit the local lifestyle. In some cases, the Japanese version ends up outshining the original, and this is certainly the case with the handkerchief.
Handkerchiefs have largely disappeared from daily life in the West, where its use is mostly limited to an accessory, like a pocket square of a man's jacket for a formal occasion, such as at a wedding. In Japan, however, the majority of the populace, from small children to the elderly, still carry a handkerchief on a regular basis.
Handkerchiefs serve multiple purposes: For drying your hands in public restrooms that don't have driers or paper towels, for blotting sweat off your face in the humid summer months, or for resting on your lap while eating a boxed lunch. In a pinch, they can cover your head during a rain shower, or be used for wiping off a wet bicycle seat when the rain has stopped.