How do I include the sidebar (with Widgets) in a custom theme?

Your issue is the dynamic sidebar being called is not matching the ID of what is registered.

Matching:

 <?php if ( is_active_sidebar( 'sidebar-widgets' ) ) : ?>
        <div id="secondary" class="widget-area" role="complementary">
            <?php dynamic_sidebar( 'sidebar-widgets' ); ?>
        </div><!-- #secondary -->
    <?php endif; ?>

Register Sidebar

if (function_exists('register_sidebar')) {
      register_sidebar(array(
        'name' => 'Sidebar Widgets',
        'id'   => 'sidebar-widgets',
        'description'   => 'These are widgets for the sidebar.',
        'before_widget' => '<div id="%1$s" class="widget %2$s">',
        'after_widget'  => '</div>',
        'before_title'  => '<h2>',
        'after_title'   => '</h2>'
    ));
}

<?php get_sidebar(); ?> tells your template to use the sidebar.php file.