Check out: http://codex.wordpress.org/Template_Tags/wp_list_categories
This page may help too with what I'm going to say next: http://codex.wordpress.org/Template_Hierarchy
Your going to want to copy your current theme's index.php file, name it home.php. Then go in and replace the_content(); or the_excerpt(); with wp_list_categories();
I would also replace the post title line, which has the_title(); in it, to be a non-linking title that says "Categories" or something.
That's the basic idea... I bet there are going to be some things your going to want to change about it, such as the text size.