Mikoto and Xander have been penpals for several years. It started around the beginning of Xander's rule as King and around the time Mikoto stepped down from being Queen- which is why they can't just end this tense not-war not-peace thing their nations have going on.
As for how it started... see, Laslow had gotten a little better at dancing in front of crowds. And he had a friend who he needed to step in for. After the performance, Mikoto went up to him to start chatting with him about his performance, which was a shock because Laslow knew faces and that was The Damn Queen (former) and Laslow was like um. And then Xander showed up.
And they got to having a chat and because Reina and Laslow were the ones around theyre the ones assigned to pass off secret letters from their respective lieges once they decided on the secret penpal thing.
Siegbert is also vaguely aware of this due to being a young child at the time and being around for the performance. He's not super involved but he knows something is going on and he's hopeful!