Syncing works both ways, so the operation is done twice.
1. Bring Airbnb's calendar in
In Airbnb: Calendar → pick the listing → Availability → Sync calendars → Export
calendar. Copy the address ending in .ics.
In your dashboard: Calendar → Add an external calendar → paste, name it "Airbnb", save.
2. Send your bookings to Airbnb
Copy your own export link from the same dashboard page, then in Airbnb: Sync calendars → Import calendar → paste it.
3. Check
Block a test date on Airbnb and look at your calendar an hour later. Remember to unblock it.
About delays
iCal is not instant: Airbnb refreshes its imports on its own schedule, often several times a day. Between two refreshes a double booking stays theoretically possible on a much-wanted date — that is the limit of the method, not a setting to fix. On those periods, block manually rather than relying on the sync alone.
If it does not work
Check you have not swapped the two links, which is the most common mistake: Airbnb's link goes into your dashboard, yours goes into Airbnb.