Why do we often find a folded-up piece of tissue paper inside one of a pair of men's dress socks?

This is a hard question to answer.

The tissue in the sock is there as part of the esthetic; to add a sense of luxury to hosiery. It adds a bit of thickness to the packaging because better men's hosiery uses finer fabric yarns, which makes them thinner than the less expensive ones. And the tissue is there to add the pleasant sound of crinkling.

http://www.nylongene.com/2010/02/history-of-hosiery.html

I'd compare that with the thin card in underpants and the plastic or stiff card in shirts, they all keep the shape and stop the material from wrinkling. I also find the tissue paper between new socks avoids static cling when you first unpack them.