That KML file does not contain the actual data, but rather is a "NetworkLink" pointer to another KML file. In order to access the actual point (or other) data, you'll need to download the KML file which it points at. If you put the contents of the
<href> tag into your browser (or just click here: http://whc.unesco.org/en/list/kmz), it will download the data KML.
In this case, the downloaded file is a KMZ file, which is a zipped version of a KML. So you'll need to unzip the KMZ (with 7zip or similar) and extract the KML file which it contains. Looking at that KML, it appears to contain some invalid characters, so it may or may not require some cleanup before you can open it in QGIS and convert to a Shapefile.
FYI that everyone can get Google Earth Pro for free now. Just download it (from here) and then use GEPFREE as the license key.
It looks like you might be trying to create a shapefile of the UNESCO World Heritage List. If so, it might be easier to use one of the links on (this UNESCO web page), where they provide direct downloads of the list in KML format, XML format, and XLS (Excel spreadsheet) format.