Early men almost certainly used animal skins to keep the chill off their feet - and the rest of their bodies. One day, a housewife was tidying up the cave again, and she collected all the shreds of wool and fur that had fallen off the skins. Being of a saving disposition, she stashed the bits in a corner and carried on. The same thing happened most days, and she continued to push the loose threads into the corner.
Eventually, when the skins were pretty well bare, she threw them out and demanded her husband get some new ones. He complained (of course), but went off with his buddies, their clubs over their shoulders, to see what they could bash.
While they were away, the housewife wanted something soft for the baby to sit on, so she pulled out the large bundle of wool and settled the baby on it near the fire. Was she annoyed when the baby did what babies do, and she had to put the warm, wet, squashed wool outside the cave! The surprise came the next day, when the damp lump had become a thick, soft pad (though smelling strongly of baby).
After that, she collected the wool and fur more carefully and each time she had a bundle big enough, she sat the baby on it, producing another pad of felt, so in a fairly short time she had enough for a rug for everyone to sit on. The idea caught on, and the demand grew for loose wool - and babies, until one day a housewife with a strong sense of smell tried spilling hot water on the wool instead, and felt was born that could be used in any shape to make all sorts of large and small rugs.