I learnt a new word - yesterday.
It is a "doily" that would have been put at your head area - on the sofa. It protected your sofa from - the oils - that were on your head.
There was a "fashion" to use an oil called - Macassar.
Since the Antimacassar was washable - it worked.
They are still used today - Trains, planes, buses use them today. They are the fabric piece used to extend the life of the fabric.