What does "den" mean here?

It's like a family room. A den is an informal common room compared to a formal living room where you entertain guests.

I think study would be appropriate. I have never heard a den referred to as a family room. In my experience, it refers to the room in the house where the man of the house can go for some peace and quiet, like a library, a study--and hence the word den. Den also means the lair or home of a wild animal, and it is used in a figurative sense.

Here is a link to something described as a "Florida room". I wouldn't consider this a family room or a study.

dens can range
a persons personal space in a single room, they call it a den
could be a mass of blankets thrown over things in order to make a little space.

Talking about Florida, Devil's den here: