How to display WordPress search results?

Basically, you need to include the WordPress loop in your search.php template to loop through the search results and show them as part of the template.

Below is a very basic example from The WordPress Theme Search Template and Page Template over at ThemeShaper.

<?php
/**
 * The template for displaying Search Results pages.
 *
 * @package Shape
 * @since Shape 1.0
 */

get_header(); ?>

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

            <?php if ( have_posts() ) : ?>

                <header class="page-header">
                    <h1 class="page-title"><?php printf( __( 'Search Results for: %s', 'shape' ), '<span>' . get_search_query() . '</span>' ); ?></h1>
                </header><!-- .page-header -->

                <?php shape_content_nav( 'nav-above' ); ?>

                <?php /* Start the Loop */ ?>
                <?php while ( have_posts() ) : the_post(); ?>

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

                <?php endwhile; ?>

                <?php shape_content_nav( 'nav-below' ); ?>

            <?php else : ?>

                <?php get_template_part( 'no-results', 'search' ); ?>

            <?php endif; ?>

            </div><!-- #content .site-content -->
        </section><!-- #primary .content-area -->

<?php get_sidebar(); ?>
<?php get_footer(); ?>