Telling someone you are enthusiastic about their idea once meant you thought a god was inside you.
The Greek verb enthousiazein meant to be inspired or seized by a god. It grew from entheos, "possessed by a god," which Etymonline splits into en, "in," and theos, "god." The noun enthousiasmos named the state itself: divine inspiration.
English got the word around 1600, from French enthousiasme and directly from Late Latin enthusiasmus. At first it kept the religious sense. Etymonline dates the adjective enthusiastic to about 1600 as well, with the meaning "pertaining to possession by a deity."
Then the word turned sour. In the 1650s, Etymonline records, the Puritans gave enthusiasm a derogatory sense: excessive religious emotion based on a claim of special revelation from God. By the 1690s, enthusiastic described irrational religious delusion. To call someone an enthusiast in that era was to say they had mistaken their own feelings for a message from heaven.
Merriam-Webster dates the first known use to 1595 and says that for its first two hundred years or so the word mostly described religious fervor. The insult faded. Etymonline finds the broader meaning of intense fervor or zeal by 1716, and it became the main modern sense. For the adjective, Etymonline gives 1786 as the date for the modern meaning of intensely eager or rapturous. Etymonline also finds unenthusiastic implied in an adverb from 1785.
Notice what the word did along the way. It started as a claim about gods, became a charge against believers, and ended as the thing every job applicant promises. The god is gone, but the structure of the word still says it: en plus theos.
Etymonline also says sarcastic, from the 1690s, was formed "perhaps on the model of" enthusiastic. The guess is uncertain.