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I can't really see it, since the site is obviously busy or down at the moment. But I think that Mickey is inviting the three to his castle, so that he can tell them about his adventures before he became a King, and HOW he became a King.
Yoshi, I highly doubt he [the King] was inviting them to his Castle. If you pay attention to Kairi's facial expression, she looks rather worried. I'm guessing the King has some sort of 'quest' for them.
I read their reactions as
Riku: Amussed
Sora: Confused
Kairi: Worried
Now for my theory. what if the letter is actually an old one, somehow realting to the secret ending? or maybe it talks about something the King learned or knows about the "keyblade wars"
NO ONE knows what was on the Letter. nomura said there MAY come a time when he can reveal just WHAT the letter said, but until then he says he wishes the fans to speculate on it.