The CD picks off from the moment Kurisu learns from Okarin that in order to save Mayuri, the timeline has to be changed and she would have to be sent off to her death. This is the Chapter 10 Kurisu route, but told from Kurisu's point of view.
For the most part, we hear much of Kurisu worrying about how she's going to disappear. She then learns that her father, Dr. Nakabachi, had been working on time travel in 1996 with Feiris' father, Akiha Yukitaka, as well as someone who you will recognize as Amane Suzuha; except, since she travelled to 1975 to get the IBN 5100 for Okarin, she's approximately 39 by this point when we hear her.
Though Nakabachi at first has some selfish reasons for time travel, we later learn that he is more motivated by the loss of Suzuha and Feiris' father; Suzuha had passed away due to illness, and Feiris' father died in a car accident. Not only that, the trauma from it all had caused him to say things he wasn't supposed to say on Kurisu's birthday in 2003, and thus he had to leave her and his wife behind. He vowed to finish time travel and go back to see them once more.
All this comes from tapes that Feiris managed to get ahold of, and convinces Kurisu to realize that unless she says the words she has to say, she will regret it for the rest of her life. By the end of the CD, Kurisu opens the door and confesses her love to Okarin, just before the divergence change.
Kurisu: The CD explains why Kurisu suddenly had a change of heart and came back to see Okarin just before he changes the timeline. There's not much more you'd expect from here other than how the story is told from another point of view. However, as Kurisu doesn't have the Reading Steiner, we're not able to see any time travel divergences or effects for this CD.
Feiris: Akiha Rumiho. It turns out her father had also worked with Shouichi in the past in time travel, too. We don't hear much else, though.
Suzuha: Under the name Hashida Suzu, Suzuha decided to work with Shouichi on time travel. As she's the most familiar with time travel theories, she probably already knows Shouichi is going to fail, as it was Okarin and Kurisu that eventually develops time machines in this divergence. It's certain that in the Beta World Line, Suzuha was not involved at this level as she gets a time machine by then that allows her to travel in both directions. It's more than likely Suzuha was here to observe and see what Kurisu's father was like, more than trying to develop a time machine, as she knows it won't work.
Despite being 39-years-old, she sounds like the kind of woman you'd like to meet out there; lots of secrets, and interesting backstory. Though don't say "past", as her "past" would be in the "future" now, wouldn't it? Amazing voicework here also gives this sort of aura away.
Nakabachi: Makise Shouichi, Kurisu's father. We know that from the Chapter 11 route, Shouichi had fallen into an argument with her father, and laughs away her death when she is stabbed by accident, due to Okarin failing his first attempt at saving Kurisu. But from the CD, we learn there is more to his character, and thus we have a classic case of a father who wishes to change everything to make things right, but can't express himself properly to others in the present.
We've never seen a scenario where Okarin sees Kurisu get killed without any of the time travel parts of the plot in play, so it's not likely Shouichi intended to kill her. However, her death in the original divergence of the game was likely a major motivator for him to develop time travel and save her, only to cause World War III. Interesting how such things can drive people to become crazy, and not realize the repercussions of doing so until much, much later.