I hope someone can answer you, because I'll be interested too. I bought one of these a few weeks ago; it's still in flower, but later I may be wondering what to do next!

The thick, bamboo-like stem is the pseudobulb, and it is still leafy and making food for the rest of the plant. Even after it loses its leaves, it stores food for the rest of the plant, and shouldn't be removed until it shrivels up--maybe 2-4 years from now.
Barbarak, yours looks like it could use some more light than it has been getting, though, for flowers next season.

Thanks tug, I moved it from an east facing kitchen windowsill to take the photo. I originally bought it for the bathroom but it was too tall for the window depth. I presume they need humidity like the moth orchids.

All of the orchids that I know of do better with humidity, but "cane-stemmed" Dendrobiums, like this one, aren't dependent on it, unlike, say, Paphiopedalums.