The reason for giving a gift is different for each person but wrapping has the also the purpose of "expressing gratitude”. It might sound obvious, but when you give a gift is for celebrate something. Birthday, engagement, wedding, admission to school, graduation, employment, coming of age day, anniversary, mother's day, father's day, etc. they all are events to celebrate.
In Japan, festivity meant sacred ceremony. In the Japanese tradition of "offering to the Gods”, to prevent contamination, the gifts were placed on a mat and covered with something to protect them from the dust.
This was the starting point of Japanese wrapping tradition, although its role has changed throughout the years.
When celebrating, I think most of the people celebrate with their families, loved ones or friends. Grateful for what we have around us and we hope for a better life. Those days represent a crossroads in our life, a point where we stop and look back on our past and forward to our future.
The wrapping is important because it covers the item we chose carefully. However, it does not matter how beautiful wrapping paper is because, it may have played an important role in protecting that gift, but at the end, it will be torn down and thrown away. But one level above, by using the furoshiki, even if the item that we received is now broken and we don’t use it anymore, we will always remember the moment when we received it, wrapped in the furoshiki.