Furoshiki was originally used at a bath (Furo) and spread on a floor(Shiki). So people could wrap up their clothes. Each furoshiki had a family crest to avoid losing their cloth while bathing.
Before paper bags became popular. furoshiki was an effective tool to wrap gifts. There are rules on how to wrap or unwrap furoshiki. For example, after unwrapping a furoshiki, fold it nicely and keep it on the left side before presenting a gift. Most of rules are not strictly followed these days.
We still keep the practice, especially when we present an envelop containing a gift of money, or the time we are invited a senior's house and carry a gift to them.
Now young Japanese have started to look back those customs and represent it in a modern way. They tie furoshiki to make a backpack without a sewing machine. Or they carry a laptop in a furoshiki bag.
The culture of furoshiki is old but it is still developing.