Hi Giovannino,
it is actually the opposite. They are widely appreciated all around the world but quite underrated in their home country, which is Britain. If I hadn't known that, I would have interpreted the phrase exactly in the way you did. I guess it is a fixed phrase, as Tim suggested, but can anybody explain in other words what it means?
Does it mean that nobody knows why Depeche Mode have had such a success or that their success lies in their being British? Or what else?
Mmmm...in this sense then, it was maybe right the "mistery" meaning you mentioned before. They are a mistery, in the sense that nobody understands why they're not popular in Britain, even if it is their home country. Why on earth they are famous and appreciated all over the world and not at home? This is a mistery, isn't it?
Let's try to explain it by example:
I think that Depeche Mode are one of Britain's
best-kept secrets.
I think that Depeche Mode is a fantastic British group not well-known (inside) (or outside) of Britain
I think you could interpret it either way.
Given what you say about the group I would have stated it differently.
I think that Depeche Mode is one of the best-kept secrets inside of Britain.
or
Though Depeche Mode is well-known throughout Europe, it is one of the best-kept secrets inside the UK borders.
Other uses:
I think that the Pasta Museum near the Trevi is one of the best-kept secrets of Rome.
I think that the Pasta Museum is fantastic, but not well-known to tourists.
I think that Monteriggioni is one of the best-kept secrets of Tuscany.
I think that Monteriggioni is fantastic and not well known to tourists.
The point I was making only related to the fixed phrase "best-kept secrets".