Translator here.
I think the "weakness" is something akin to the fear tell in Part 4 of JoJo (knowers know). But, basically, it just has to be what one fears the most, something that can be used against them with maximum leverage. For example, in the first chapter MC mentions that one's weakness is also swayed by their values. So, the dude in chapter one didn't consider it important if someone knew he was guilty, but when that dude started pressing him and got closer to the truth than anyone ever had, it may be the most vulnerable he'd ever been, making him susceptible to the card.
Just a guess, though, I'm very busy so I haven't had much time to read ahead so I don't know if there's a better explanation. But basically, as far as I can tell, by telling someone something about them that they would do anything to not have used against them, you make them vulnerable to the card.