Under "pricing and availability" you can get a URL to have your calendar exported to a Google calendar or other ICAL compatible calendars, like Apple. However, that means you or your housekeeper will have to know how to go into your calendar settings and create the sync.
I've just started taking a screenshot of the Airbnb calendar for the current month and sending it to our housekeeper. If a cancellation happens or something changes, I send her a new screenshot. It's not the *best* method, but it's simple and works for us.