Here's my theory. The story Shizuku wrote is about her and someone who used to be dear to her (I don't think it's a girlfriend/boyfriend) and something tragic happened to them, and Shizuku feels guilty about it that's why she became anti-social to prevent herself from getting hurt or hurting someone anymore. And Kaori is someone who knows about her past (or came to that conclusion herself after reading the story) so she wanted to open Shizuku's heart again (I can't guess the reason thought, there's something fishy about Kaori in the first place) that's why she suggested to date in a summer vaction's time. But that'd probably end tragically, because Shizuku said "The first and last time I saw her was that summer".
Here's my theory. The story Shizuku wrote is about her and someone who used to be dear to her (I don't think it's a girlfriend/boyfriend) and something tragic happened to them, and Shizuku feels guilty about it that's why she became anti-social to prevent herself from getting hurt or hurting someone anymore. And Kaori is someone who knows about her past (or came to that conclusion herself after reading the story) so she wanted to open Shizuku's heart again (I can't guess the reason thought, there's something fishy about Kaori in the first place) that's why she suggested to date in a summer vaction's time. But that'd probably end tragically, because Shizuku said "The first and last time I saw her was that summer".