How can I make a HTML5 game like this?

Correct me if I'm wrong, but as far as I can see that game is just static images with text and buttons overlaid. As such, Unity would be massive overkill. You could build it quite simply in basic HTML, CSS and JS.
 
Most Web apps were done in flash or silverlight, both are being fazed out in favour of HTML5 which works alongside js and css.

Google for a 'download whole webpage' plugin and you will be able to download all the js and css for that site.
 
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