놀다 for adults means hanging out and enjoying yourself, not children's play. -어서 놀다 puts the getting-there first, and most lines here pair it with 나가서 "go out and hang out" or 모여서 "get together and hang out". The "because" reading of -어서 also appears: 사준다고 해서 놀려고요 "they offered to treat me, so we're meeting up". Saying 놀다 about a grown adult is completely normal in Korean, unlike English "play".