The espadrille (or alpargata) originated as early as 4000 years ago on the Iberian Peninsula. For much of their history, they were worn by peasants around the Pyrenees. Between the 1200s and 1800s, espadrilles were also worn by soldiers and mine workers. Over time, a variety of regional styles developed. While all share braided jute or hemp rope soles, which provide protection from extreme temperatures and humidity, different regions created unique designs for the linen uppers and laces.
It was not until about the 1940s-1950s that style became elevated to higher fashion. In 1940, just after the end of the Spanish Civil War, a shoe store called La Manual Alpargatera was founded in Barcelona. Leather was in short supply, so the store focused on making the traditional peasant shoe for a broader audience, and they were very successful. Still in operation today, the store makes espadrilles for everyone from Salvador Dali to the Catalonian police to Pope John Paul II. (Incidentally, I bought these shoes at that same store.)
The rise of the style can also be credited in part to Edward VIII of England. At around the same time as the store was founded, Edward discovered espadrilles in Biarritz, France and thereafter wore them frequently, exposing the elite British society to the style. Early American movie stars like Lauren Bacall and Grace Kelly then popularized the style in the US. The shoe achieved real fashion legitimacy in the early 1970s when Yves Saint Laurent created the espadrille with the heel.