Content Query Web Part error “Unable to display this Web Part. To troubleshoot the problem, open this Web page in a Microsoft SharePoint Foundation-compatible HTML editor such as Microsoft SharePoint Designer. If the problem persists, contact your Web server administrator.”

Well this error occurs when you have not configured AAM (Alternet Access Mapping)for the particular web application. Suppose if the internal URL is http://server:81 and you are accessing the URL out of the server by mapping the DNS, then you will face this issue in the web part. SO, to resolve this issue just configure the public URL for the http://server:81 to like http://www.sample.com/.net etc.