Wet hair is wound and secured in curlers (and the vintage types work best), dried either naturally or with a dryer, and then brushed and/or combed into place. A style done this way can last a much longer that with hot rollers or irons, and can be less damaging. This picture is from a vintage late 1960s or early 70s package of hair rollers.
These rollers are Wil-hold plastic brush rollers, probably some of the most popular curlers at the time for salon use but also for at home (and this package was intended for home users). They occasionally appear on eBay and in my opinion they are some of the best rollers ever made. (More about Wil-hold rollers later).