Is vacation rental site Airbnb illegal?

Yes it's illegal. There has never been any dispute on this matter. In practice it's a relatively unenforced matter.

How? If I understand how Airbnb works then they don't run the rooms, just the booking. So what's illegal about taking appointments?

You know, there's more than one set of laws in the world.

Airbnb is legal. It's their users who may be breaking the law. Like Craigslist and users who aren't paying use taxes on the stuff they buy.