c. 1200, "a way, a road, space for passage," from Old French rute "road, way, path" (12c.), from Latin rupta (via) "(a road) opened by force," broken or cut through a forest, etc., from rupta, fem. past participle of rumpere "to break" (see rupture (n.)).
The sense of "fixed or regular course for carrying things" (originally and for long especially postal, as in mail route) is from 1792, an extension of the meaning "customary path of animals" (early 15c.) itself later extended to sales, collections, delivery of milk or newspapers, etc. OED says the pronunciation that rhymes with "stout" appeared early 19c.
1890, of a railroad ticket, "mark for use on a certain route," from route (n.). The meaning "direct (an electrical signal, phone call, etc.) over a particular circuit or to a particular location" is by 1948. Related: Routed; routing; routeing (1881).