Hello.
To achieve similar effect like on site you refer to (double-horizontal menu), I would recommend you to use one CSS List menu web part underneath second CSS List menu web part. First of them (on top) will be horizontal menu for first level items. You can see how to style CSS List menu to be horizontal
here. Second menu will be the same. Only difference will be in
Path property, which will be set for pages under currently selected page. This can be done via
path expression.
Best Regards,
Radek Macalik