There's gentle irony in the statement. It's something you can say about your fame, especially if you're unknown in your own country. The point is, it doesn't necessarily have to be the truth - it's used jokingly to excuse a lack of fame at home. This is now a well-known catchphrase, probably thanks to the song. "I'm in a rock band." "Really? I've never heard of you." "Well, I'm big in Japan." Of course, very few Westerners would bother going to Japan to verify this. (My Korean friend also pointed out that Japan is geographically a "smaller" country, so you could be famous, but your fame is very limited.)

Андрей, According to the lead singer Marian Gold, who wrote the song. "There was a fellow, who sold stuff of independent bands from US and UK. I bought the album of a British band named Big In Japan. "Big in Japan" meant so much as if you are nothing in your environs, you can be big somewhere else. You can be a KING in another world. And if you aren't that, you can tell it at home. Japan is so far away. Anyway that statement fit perfectly to the story of the couple of lovers." He is speaking about the fact that some english speaking rock bands who are no longer popular at home can still be popular in Japan. The Russian translation 'Властелин мира' i.e. 'Lord of the World' seems a bit exagerrated. Maybe you can think of a better translation for "Big in Japan". How about "Звезда в Японии". Star in Japan Or simply "Большой в Японии" Big in Japan What do you think?