Rayon is made from cellulose extracted from wood pulp but it's not, technically, a natural fibre because a high degree of chemical processing is required. It was called artificial silk but is now largely referred to as viscose.
Any materials made from cellulose have a high level of water absorbtion so pure Rayon material was best dry cleaned as rough handling in a washing machine could damage it.
It was popular back then because it dyed easily, draped well and was easy to work with at home. It just fell from popularity as it was superseded by more advanced fabrics that were easier to clean, keeping their shape in a washing machine.
Viscose is still used today in blends to modify the properties and give a different material than either of the individual components. I expect rayon crepe is just another type of one of these fabrics.