The best-known story about sabotage is the one that is not true. Angry workers, it goes, threw their wooden shoes into the machines. Merriam-Webster says plainly that "there is no evidence for such an etymology." Wikipedia calls one popular version, set among poor workers in Liège, Belgium, inaccurate.
The real path runs through noise and clumsiness. A sabot is a French wooden shoe. The verb saboter could mean to make a clatter in them. In 1808 D'Hautel's dictionary of low French recorded saboter as "make noise with sabots" and also as "bungle" or "hustle." Walking loudly in clogs became a picture of doing anything clumsily. French speakers used it for all kinds of bungling, including playing music badly.
Merriam-Webster notes one open question. It is unclear whether "clatter" and "botch" grew one from the other or were separate new verbs made from sabot. Either way, neither sense involves throwing a shoe at anything.
The labor meaning came in the 1890s. In 1897 the anarchist and syndicalist Émile Pouget wrote of saboter un travail, "sabotaging a piece of work," in his paper Le Père Peinard. The idea was that workers need not strike. They could stay on the job and do it badly, so the employer lost money. Pouget later wrote a book called Le Sabotage, published in 1911.
English borrowed the word in that sense. Etymonline finds it as a French term in English from 1903 and as an English noun by 1907. One early English description explains the tactic: the worker would "remain on and do his work badly, so as to annoy his employer's customers." In the United States the Industrial Workers of the World promoted it too. Elizabeth Gurley Flynn's IWW pamphlet defined it as "the conscious withdrawal of the workers' industrial efficiency."
The violent meaning is later. During World War I sabotage expanded to mean damage done in secret to disrupt an enemy. That is the sense most people have in mind today, and it probably made the clog story feel right. A shoe in the gears is a vivid image of wrecking a machine. The word itself began with something quieter: a worker in clogs doing a deliberately poor job.