What story does "Love Letter" tell?
The game is a bit abstract, when it comes to the theme connection with mechanics, but I don't think it's quite that confusing.
Quoting from Premium Edition Rules (emphasis added)

In the wake of the arrest of Queen Marianna for high treason, none was more heartbroken than her daughter, Princess Annette. Suitors throughout the City-State of Tempest sought to ease Annette’s sorrow by courting her, seeking to bring some joy into her life.
You are one of these suitors, trying to get your love letter to the Princess. Unfortunately, she has locked herself in the palace, so you must rely on intermediaries to carry your message.
During the game, you hold one secret card in your hand. This is who currently carries your message of love for the Princess. Make sure that the person closest to the Princess holds your love letter at the end of the day, so it reaches her first!

The card you hold out of your turn would represent who's currently holding your love letter. At the end of the round, the closer that closest person to the princess is the one to deliver that suitors letter first. Apparently its first come, first served, but lets not dwell on that.
On your turn you have a chance to have one character exert their influence over the other suitors or at least the people happening to hold their letters, while the other gets your letter passed to them.
How exactly are you getting all these people, all the way to the king, to help you out with this? We don't really know, maybe you're just so sweet with your words.
As for the princess, you have somehow managed to get the letter directly at her hands. However, she doesn't read it immediately but instead...

[...] Obviously, you want the princess to carry your letter. However, she is self-conscious about matters of the heart, and if confronted, will toss your letter in the fire and deny looking at any correspondence.
If you discard the Princess — no matter how or why — she has tossed your letter into the fire. [...]