Edit a post from frontend. post_tags get saved, but not separated

I’m building a CRM-Theme for WordPress. So i post, edit and delete posts from the Frontend. I’m on a good way to resolve all problems, i think. But there is one failure. At the index there is a “edit” and “delete” button. If i click on “edit” I’m getting to a form. I can change the post_tags there. Changes will be saved, but the single words don’t get separated. All words in the input-field are one huge tag – i can see it in a widget at the sidebar. But i use separators… If you have 2 minutes:

Here (in the index.php) i enable to delete a post:

if( 'POST' == $_SERVER['REQUEST_METHOD'] ) {
   set_query_var( 'postid1', $_POST['postid'] );     /*  set the "postid" value from the delete button of the post we choose to delete into "postid1" */
   wp_delete_post( get_query_var( 'postid1'), true );  /* delete the post we choosed */
};
get_header(); ?>

In the loop of the index you can find the following snippet. First form is for edit, second form for delete:

<form class="edit" action="<?php echo esc_url( home_url( '/' ) ); ?>editieren-formular/" method="post">
    <input type="hidden" name="postid" value="<?php the_ID(); ?>" /> <?php /* get the post ID into "postid" and later pass it to "editieren-formular.php" */ ?>
    <input type="submit" value="Edit" />
</form>
<form class="delete" action="" method="post">
    <input type="hidden" name="postid" value="<?php the_ID(); ?>" /> <?php /* get the post ID into "postid" and later delete the post */ ?>
    <input type="submit"  value="Delete" />
</form>

Everything fine. Now the editieren-formular.php, which is a page-template for a site, that is called “Editiere Formular”. “editieren-formular” is the slug.

<?php
/*
Template Name: Editieren Formular
*/

if( 'POST' == $_SERVER['REQUEST_METHOD'] && !empty( $_POST['action'] ) &&  $_POST['action'] == "edit_post" && isset($_POST['postid'])) {
    $post_to_edit = array();
    $post_to_edit = get_post($_POST['postid']);

    /* these are the fields that we are editing in the form below. */
    $title = $_POST['title'];
    $description = $_POST['description'];
    $vorname = $_POST['vorname'];
    $name = $_POST['name'];

    /* this code will save the title and description into the post_to_edit array */
    $post_to_edit->post_title = $title;
    $post_to_edit->post_content = $description;

    /* honestly i can't really remember why i added this code but it is a must */
    $pid = wp_update_post($post_to_edit);

    /* save taxonomies: post ID, form name, taxonomy name, if it appends(true) or rewrite(false) */
    wp_set_post_terms($pid, array($_POST['cat']),'firmen',false);
    wp_set_post_terms($pid, array($_POST['post_tags']),'post_tag',false);

    //UPDATE CUSTOM FIELDS WITH THE NEW INFO
    //CHANGE TO YOUR CUSTOM FIELDS AND ADD AS MANY AS YOU NEED

    update_post_meta($pid, 'vorname', $vorname);
    update_post_meta($pid, 'name', $name);

    //REDIRECT USER WHERE EVER YOU WANT AFTER DONE EDITING
    wp_redirect( 'http://crm-wordpress-theme.wellseo.de' );

...

} // END THE IF STATEMENT THAT STARTED THE WHOLE FORM


get_header(); 

$post_to_edit = get_post($_POST['postid']); 
$terms = get_the_terms($post_to_edit->ID, 'firmen'); 
$tags = strip_tags( get_the_term_list( $post_to_edit->ID, 'post_tag', '', ', ', '' ) ); 

<?php $term_name = strip_tags( get_the_term_list( $post_to_edit->ID, 'category', '', ', ', '' ) ); ?> <!-- get the category name of this post -->
<?php $term_obj = get_term_by('name', $term_name, 'category'); ?> <!-- get the current term object -->
<?php $term_id = $term_obj->term_id ;?> <!-- get this post's term id -->
<?php $args = array(
    'selected' => $term_id,
    'name' => 'cat',
    'class' => 'postform',
    'tab_index' => 10,
    'depth' => 2,
    'hierarchical' => 1,
    'hide_empty' => false );  /* array for wp_dropdown_category to display with the current post category selected by default */ ?>

<div id="content" role="main">

<form id="edit_post" name="edit_post" method="post" action="" enctype="multipart/form-data">
    <fieldset name="title">
        <label for="title">Title:</label><br />
        <input type="text" id="title" value="<?php echo $post_to_edit->post_title; ?>" tabindex="5" name="title" />
    </fieldset>

    <fieldset id="category">
        <label for="cat" ><?php wp_dropdown_categories( $args ); ?></label>
    </fieldset>

    <fieldset class="content">
        <label for="description">Discount Description:</label><br />
        <textarea id="description"  tabindex="15" name="description"><?php echo $post_to_edit->post_content; ?></textarea>
    </fieldset>

<!-- BELOW ARE THE CUSTOM FIELDS -->

    <fieldset class="vorname">
        <label for="vorname">Vorname:</label><br />
        <input type="text" value="<?php echo get_post_meta($post_to_edit->ID,'vorname', true); ?>" id="vorname" tabindex="20" name="vorname" />
    </fieldset>

    <fieldset class="name">
        <label for="name">Name:</label><br />
        <input type="text" value="<?php echo get_post_meta($post_to_edit->ID,'name', true); ?>" id="name" tabindex="20" name="name" />
    </fieldset>

    <fieldset class="tags">
        <label for="post_tags">Additional Keywords (comma separated):</label><br />
        <input type="text" value="<?php echo $tags; ?>" tabindex="35" name="post_tags" id="post_tags" />
    </fieldset>

    <fieldset class="submit">
        <input type="submit" value="Speichern" tabindex="40" id="submit" name="submit" />
    </fieldset>
    <input type="hidden" name="postid" value="<?php echo $post_to_edit->ID; ?>" />
    <input type="hidden" name="action" value="edit_post" />
    <input type="hidden" name="change_cat" value="" />
    <?php // wp_nonce_field( 'new-post' ); ?>
</form>
</div><!-- #content -->

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