How does Beatrice remember about Subaru? In episode 5 of the anime Subaru experiences the first Return by Death at Roswaal's mansion: after he runs away from his bedroom, he meets Beatrice, who remembers him even if that should be the first time they meet in that "loop". How can she remember him?
Because the "save point" for that part of the story is not the point when Subaru wakes up for the first time, but when he wakes up for the third time. That's also when he first meets Ram and Rem. (His first waking is before the opening in episode 4, while his third is right after the opening.*)

*At the start of episode four 


   Subaru wakes up, notes that it's an unfamiliar ceiling, walks around a bit, finds the library, talks with Beatrice then falls unconcious because Beatrice suck some of his mana.