Add posts of a certain category to a page

Create a page template and use wp_query to target the category you want to display

$args = array('category_name' => 'your category name',);

WP_Query will help you with all sorts of options for what posts are returned and get displayed by your loop.

Here is an example of a page template, (this is based on the _s theme)

<?php
/**
 * Template Name: Your template name
 *
 * @package _s
 */

get_header(); ?>

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

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

        <header class="entry-header">
            <?php the_title( '<h1 class="entry-title">', '</h1>' ); ?>
            <div class="entry-meta">
            </div><!-- .entry-meta -->
        </header><!-- .page-header -->

        <?php 
         $args = array(
         'category_name' => 'your category name',        
         );             
        ?>

        <?php
         $the_query = new WP_Query($args); 
         while($the_query->have_posts()) : $the_query->the_post(); 
        ?>

            <?php
                /* Include the Post-Format-specific template for the content.
                 * If you want to override this in a child theme, then include a file
                 * called content-___.php (where ___ is the Post Format name) and that    will be used instead.
                 */
                get_template_part( 'content', get_post_format() );
            ?>

        <?php endwhile; ?>

    <?php else : ?>

        <?php get_template_part( 'content', 'none' ); ?>

    <?php endif; ?>

    </main><!-- #main -->
</section><!-- #primary -->

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

You can also generate your wp_query making things a bit easier.