When a user creates a post (pending), send a confirmation link that allows them to publish

You need a function that hook into post status transitions and send and email containing a link with some verification variables and a function that takes that variables, check them and if all ok update posts status.

Following code I think is self-explanatory, and comments give additional help:

add_action('new_to_pending', 'send_approve_link');
add_action('draft_to_pending', 'send_approve_link');
add_action('auto-draft_to_pending', 'send_approve_link');
add_action('init', 'approve_post');

function set_html_content_type() { return 'text/html'; }

function send_approve_link( $post ) {
  // get author of post
  $author = new WP_User($post->post_author);
  // create a key for security check and save as meta
  $check = md5( $author->user_email . $post->ID  );
  update_post_meta( $post->ID, '_approve_key', $check);
  // set the content for the email
  $content="<p>Please click following link to allow the publishing of your post: &quot;";
  $content .= esc_html( get_the_title($post->ID) ) . '&quot;</p>';
  // create an url vor verify link with some variables: key, post id, and email
  $url = add_query_arg( array('email'=>$author->user_email,'approve'=>$post->ID), site_url() );
  $content .= '<p><a href="' . $url . '">Click to Confirm</a></p>';
  $from = get_bloginfo('name');
  $from_email = get_option('admin_email');
  $headers="From: " . $from . ' <' . $from_email . '>' . "\r\n";
  // sending email in html format
  add_filter( 'wp_mail_content_type', 'set_html_content_type' );
  wp_mail( $author->user_email, 'Confirm your Post', $content, $headers);
  remove_filter( 'wp_mail_content_type', 'set_html_content_type' );
}

function approve_post( ) {
  if ( isset($_GET['approve']) && isset($_GET['email']) ) {
    // have we all variable needed?
    if ( empty($_GET['approve']) || ! filter_var($_GET['email'], FILTER_VALIDATE_EMAIL) ) return;
    // get post
    $post = get_post($_GET['approve']);
    // this post has an author?
    if ( ! isset($post->post_author) ) return;
    // is the post already published?
    if ( $post->post_status == 'publish' ) {
      wp_redirect( get_permalink($post->ID) );
      exit();
    } elseif ( $post->post_status != 'pending' ) { // was the post deleted by admin?
      return;
    }
    // get the author
    $author = new WP_User($post->post_author);
    // check author variable
    if ( ! isset($author->user_email) ) return;
    // verify email
    if ( $_GET['email'] != $author->user_email ) return;
    // verify key
    $key = get_post_meta( $post->ID, '_approve_key', true);
    if ( $key != md5( $author->user_email . $post->ID ) ) return;
    // ok, update status
    $post_data = array('ID' => $post->ID, 'post_status' => 'publish');
    $update = wp_update_post($post_data);
    // update failed...
    if ( ! $update ) return;
    // delete verify key
    delete_post_meta($post->ID, '_approve_key');
    // work finished, view the post
    wp_redirect( get_permalink($post->ID) );
    exit();
  }
}

Thats, all. Please note code is untested.