This is pretty much answered by Chika herself in this episode. What she says is as follows (emphasis mine):
From the English dub (I'm using brackets to indicate where I'm uncertain what she's saying, as she's babbling and there are no subtitles. It's close enough anyway.):
I don't want anyone to [come] take you away, Kaguya! And I don't want you to get expelled! Please, don't leave me, I love you more than any [boyfriend/boy can]! I love you sooooo much!
From the English subtitles on the Japanese dub:
I don't want anyone to have you, Kaguya! I don't want you expelled! No, no, don't go away! I'm crazy about you! I love you!
The expulsion part gets into something not mentioned by the question, namely that Shirogane threatened to tell a teacher if Kaguya goes on her date. That in of itself is a reason for Chika to so passionately interfere in this manner. That isn't all, of course...
So, what does Chika care? It seems straight forward to me. It's perfectly clear that as Kaguya's close friend, she loves Kaguya. She directly states that here, and many times elsewhere. For instance, the relationship charts at the beginning states that Fujiwara "Super totally ✩ loves her ♡", which goes a long way in describing the depths of her feelings. In this episode, Kaguya says she's really going to go unless she gets a love confession, and that's what she gets here from Chika (There's the funny joke we're looking for! The timing is definitely too perfect for that not to be what the author was going for.). Even at the beginning of Chika and Kaguya's conversation, Chika is fidgeting a lot, so she's worrying about it, and she keeps asking Kaguya if she's really going.
This is not the only time Fujiwara is overprotective of Kaguya when it comes to relationships. In chapter 113, after saying she got hit on at a school festival, Fujiwara warns Kaguya, saying,
You should never go to their culture festival, Kaguya! You'd really get harassed! (Because you're so cute!)
Additionally, Chika has been shown to have a selfish personality. For instance, when they learn that she and Ishigami share a birthday, she becomes extremely upset, because she believes this will result in her having a subpar birthday party. Someone who can use the kind of logic she uses to deduce her birthday revelry will suffer from sharing the day with someone else can surely determine what the result will be should Kaguya actually find a boyfriend. Kaguya's time would be monopolized by this boyfriend, leaving no time for Chika herself, and this would be a greatly undesirable outcome.
I see questions all the time about why this character or that character has any sort of invested interest in another character's romances, but they always strike me as odd, as if there is some belief that romantic relationships are the only kind of relationships that have importance. Yet, there is nothing unusual about someone getting jealous of their friend's romantic partners, particularly if they're as close as Kaguya and Chika. Seeing all the hilariously selfish stuff I've seen Chika pull in the manga, I can fairly confidently say she is the kind of person who would get jealous if Kaguya got a boyfriend and spent less time with her. Though Chika has also shown incredible selflessness, such as when she's teaching Shirogane. It's thus difficult to say what she would or wouldn't do. Though here, with the added motivation of keeping Kaguya from getting expelled, she tries to dissuade her.
This is quite a heavy spoiler coming from the end of volume 22, so I highly recommend not looking at it, however it supports my interpretation of Chika so well I couldn't help but want to include it.
From the preview description for volume 23: "Will Miyuki replace Chika in Kaguya's heart? When dense Chika belatedly finds out they're dating, she certainly thinks so."
Volume 23 unsurprisingly, has Chika initially being surprised, but very accepting and supportive of Kaguya's relationship with Shirogane when Kaguya initially tells her about it in Battle 222: Kaguya Shinomiya's Impossible Demand: "Mount Horai's Branch of Pearls". Here's a snippet of the conversation between Kaguya and Chika, with emphasis being mine here, as it supports my initial characterizations of their relationship and how the story portrays it:
"Why didn't you tell me before?!"
"Because... You're special. And now someone else is special to me too. I feel like I've betrayed you."
"You can have as many special people in your life as you want! I don't mind!"
However, true to her capricious nature and joker role in the story, Chika completely flips her position in the next chapter and is very upset by Kaguya and Shirogane's relationship. Her irritation initially sparks from seeing Miko and Ishigami act like it was obvious Kaguya and Shirogane were in love. Here are some highlights from Battle 223: Chika Fujiwara Won't Admit it. Emphasis is again mine, as some of this pretty much confirms my interpretation of the conversation in the pilot as being how the author intended it to be interpreted.
"Arghhhh! What is happening?! I can't take it anymore! You stupid, dumb, moronic idiots! I can't accept this relationship! I can't stand this lovey-dovey atmosphere either! My precious Kaguya has been defiled by Shirogane. This is... cuckoldry!"
"But I will not acknowledge this couple! Kaguya is mine!
When pressed on her change of heart, this is Chika's explanation:
Kaguya: "But when I told you, you said you were happy for me..."
Chika: "Changing your mind is a sign of maturity. When I saw you two together now and was truly honest with myself, I realized I can't accept this. I will crush and destroy your relationship, whatever it takes!"