Show About and Contact Us page when they’re clicked in the top menu.

You need to include a loop to retrieve the page content. Something like:

<div id="primary" class="site-content">
    <div id="content" role="main">

     //BELOW HERE
      <?php while ( have_posts() ) : the_post(); ?>

        <?php get_template_part( 'content', 'page' ); ?>

        <?php comments_template( '', true ); ?>

      <?php endwhile; // end of the loop. ?>

     //ABOVE HERE

      </div><!-- #content -->
</div><!-- #primary -->

More on Page Templates:

Possible Tutorial Starting Point

And Page Templates from the WP Codex on Page templates:

page.php — If a specialized template that includes the page’s ID is not found, WordPress looks for and uses the theme’s default page
template.

So if you haven’t assigned anything in the admin, and no more specialized templates exist, your page.php file is being called.