It's a technical issue.
AirBNB doesn't reply with a propper response when the iCal is requested by a website instead of a browser. If you fetch the iCal data with file_get_contents($url), you get a 403 error. But if it's requested with curl and you set the user agent it gives a propper response.
In PHP this is a working sample:
<?php
$ch = curl_init();
// set your iCal url
curl_setopt($ch, CURLOPT_URL, "https://www.airbnb.nl/calendar/ical/xxxxxxx.ics?s=xxxxxx");
//return the transfer as a string, here I fake the user agent, AirBNB thinks it's a request from Firefox / mozilla
curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($ch,CURLOPT_USERAGENT,'Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.8.1.13) Gecko/20080311 Firefox/2.0.0.13');
// $output contains the output string
$output = curl_exec($ch);
// close curl resource to free up system resources
curl_close($ch);
echo $output;
?>