She sees a kindred spirit. Garen lives the life Katarina does, and in some sense they both can tell that they are sick of it. They have really high expectations for set for them. In the heat of battle they don't have any room for error or else moral will take a huge dive. A lot of weight they have to carry (n00b team sigh).
They're also equals in battle, something Katarina has never come across. In my headcanon they're the strongest fighters in Runeterra, battle-wise. After so many battles against one another there is an understanding, even a "relationship" that they hold. They probably always think of new ways to corner the other, try and triumph, but this sort of challenge is something they fear is once in a lifetime. Once they are victorious, there will be no other quite like their old rival.
Also, there is the irony that in two completely different societies, under two completely different ideologies, they are just a reflection of one another. Thats something fantastic and profound to Katarina, and to Garen. They aren't fools who just swallow the propaganda of their respective nations, mostly because of their rivalry which changed how they think. They want to know more about the other because through their clashes they have recognized how similar, how close they really are. The only one who truly can relate to their personal struggles and thoughts. Someone to be honest to, someone who would understand. Garen wants this as much as Katarina wants this.They just want to talk, in a moment of peace and quiet, and see how deep their similarities are. But reality is harsh, and all they can really do is speak with their weapons.
(Hint: They perfect matcharino)
But its not just Romeo and Juliet.