More or less, Rem and Subaru are not separated in the carriage. Betelgeuse crucifies both of them with his unseen hands, and challenges them to his game: He's going to torture one of them, and for as long as the person being tortured does not make a sound, the other is spared. He starts with Rem and removes all of her fingernails and toenails, but she doesn't make a sound. By this point Subaru is kind of sane again and witnesses it. After this he starts removing Rem's skin. Around the point that a quarter of her skin is removed she finally makes a noise and starts pleading to Betelgeuse to continue torturing her instead of moving on to Subaru, but he declines.
So he moves on to Subaru, and after his first fingernail is removed he makes a sound. Internally he wants to kill himself for not giving Rem more time to recover, because now she has to be tortured again. This continues until Rem has all of her skin removed, her eyes gouged out, and her horn dug out from her skull before she finally dies. This is where Subaru snaps like in the anime, except Betelgeuse kills him at this point rather than leaving him behind. After this he wakes up and the ending of this episode happens.