Themedemo is a demo theme and you should use it only if you want to learn how Drupal theme variables work.
It has no CSS at all and all it's variables inside *tpl.php files are printed as a list.
There are also some examples inside template.php where some custom variables are inserting into each template.
How to use it
Install a Drupal site with Standard profile. Core's Bartik theme is the default theme.
Add some nodes, taxonomy terms, users etc.
Set the Themedemo as default. Browse through the new content you have created so you can see what is the theme variables output ($page, $node, $user etc).
Who should use this theme
You should use this theme if you are a beginner Drupal theme developer and you want to clarify theme variables.
You should also try Theme Developer module.