early 15c., "gentleness, lightness," from Old French facilité "easiness, ease," from Latin facilitatem (nominative facilitas) "easiness, ease, fluency, willingness," from facilis "easy to do," from facere "to do" (from PIE root *dhe- "to set, put"). First in a medical book:
If it be nede forto smyte [the head] wiþ a malle, be it done with esynez or facilite [transl. Guy de Chauliac's "Grande Chirurgie"]
Its sense in English expanded to "opportunity" (1510s), to "aptitude, ease, quality of being easily done" (1530s). Meaning "place for doing something" which makes the word so beloved of journalists and fuzzy writers, first recorded 1872, via notion of "physical means by which (something) can be easily done."