You can do this in two ways.
Method 1:
Call it using the same method that Mark Baker Suggested like using curly braces. Curly braces are used to explicitly specify the end of a variable name.
$result->HotelListResponse->HotelList->{'@activePropertyCount'};
You can refer this link for more info http://php.net/manual/en/language.types.string.php#language.types.string.parsing.complex
Method 2:
Assign @activePropertyCount
to some variable. You can do like :
$var = '@activePropertyCount';
Then you can get the result from :
$result->HotelListResponse->HotelList->$var;
Hope this helps you :)