The general intensifier really arrived at its current meaning around 1561. The word in its original meaning of “actually”, from real + -ly, is attested from the 1450s. Real is one of the many words borrowed into English from Norman French in the 1300s, in this case from reel, meaning “actual”. Reel descends straightforwardly from Latin reālis, also meaning “actual”, which derives from earlier Latin rēs, meaning “thing” or “matter”.
Really’s gradual shift from meaning “actually” to meaning “very” is surprisingly precedented! The “actually” sense of truly (Old English trēowe, meaning “true”) goes back over a thousand years, but its recent meaning shift toward “very” is not even acknowledged by the OED yet. (It is acknowledged in the other dictionaries I checked.) Even the canonical general intensifier very (Latin vērus, meaning “true”) shifts in meaning from “actually” in 1200-1500 to, uh, “very” after 1550. Literally (Latin littera, meaning “letter”) in the process of shifting from “actually” to “very” is in very good company.