From social stirring to feeling
The word is first recorded in the 1570s meaning "a stirring, agitation," often of a social or public kind, from Middle French émotion and Old French emouvoir ("to stir up"). It traces further back to the Latin emovere, "to move out, remove, agitate," built from ex- ("out") and movere ("to move"). Its modern sense of "strong feeling" is first recorded in the 1650s, and the word broadened to cover any feeling, strong or mild, by 1808.
A word about movement
The Latin components are worth sitting with: ex- marks a forceful separation of one thing from another, while movere describes something being physically shifted from one place to another. Put together, the word's history describes emotion as a kind of displacement — something set in one place now being pushed toward somewhere else.