I remember posting about how this didn’t make sense way back in 2015. I finally found an interview where Abrams said that when BB8 first meets R2, he asked him if he knows anything about Luke’s whereabouts. R2 heard the question while in low power mode, and slowly wakes up over the course of the movie. Upon fully booting up, he searches his databanks and finds the map, which he unknowingly downloaded when he copied a bunch of files from the Death Stars computers.
Apparently that was Abrams’s intention and there’s absolutely nothing on screen that communicates any of it to the audience.