View source shows that your categories are there, but the inline styling of the "advancedcategories" mod has it all hidden. Apparently whatever is supposed to make the categories visible is not working right, but we would need to be familiar with the javascript and/or CSS it uses to know why. I notice references to "includes/templates/template_default/css/slide-out-menu-new.css" and "includes/templates/template_default/jscript/slide-out-menu-new.js"; if either of those is missing or corrupt, it would not work.
Are these files supposed to go in /template_default/, or in your custom template folders? If they are supposed to be in your template, they are not being found and are being called from /template_default/ where they do not exist either.