Drupal 8, include a subtemplate in a form
Most likely you are looking at adding to the "render array" of the page (or form). A render array can refer to a "type" or Render Element, and that type can render into a template.

The scenario of rendering after form submission on the same form can be done within the Form itself by setting the form state to rebuild.

  public function submitForm(array $form, FormStateInterface $form_state) {

    // Set the rebuild state to TRUE to persist $form_state when the form is reloaded.
    $form_state-&gt;setRebuild();
    $form_state-&gt;set('display_extra', TRUE);
  }

  public function buildForm(array $form, FormStateInterface $form_State) {

    // Conditionally display something from form state.
    if ($form_state-&gt;has('display_extra') &amp;&amp; $form_state-&gt;get('display_extra')) {
      // Structure the render array here including setting up a custom RenderElement ('#type'), which has a custom template defined in hook_theme().
      $form['display_something'] = [];

    }

    // Rest of form...

    return $form;
  }