How to exclude one category

You can use “category__not_in” array parameter. Also, I’ve modified your code just a bit, you should use wordpress best practice which is “if -> while” to display the posts.
Please try these code below as I haven’t tested it yet.

<?php
// Default arguments
$args = array(
    'posts_per_page'    => 6, // How many items to display
    'post__not_in'      => array( get_the_ID() ), // Exclude current post
    'no_found_rows'     => true, // We don't ned pagination so this speeds up the query
    'orderby'           => 'rand',
    'category__not_in'  => array( 1 )

);

// Check for current post category and add tax_query to the query arguments
$cats = wp_get_post_terms( get_the_ID(), 'category' ); 
$cats_ids = array();  
foreach( $cats as $wpex_related_cat ) {
    $cats_ids[] = $wpex_related_cat->term_id; 
}
if ( ! empty( $cats_ids ) ) {
    $args['category__in'] = $cats_ids;
}

// Query posts
$wpex_query = new WP_Query( $args );

// Loop through posts
if( $wpex_query->have_posts() ) :
while( $wpex_query->have_posts() ) : $wpex_query->the_post(); ?>

<div class="weitere_interessante_artikel_container">
<div class="weitere-artikel-pic-wrapper">
<a href="https://wordpress.stackexchange.com/questions/242083/<?php the_permalink() ?>"><?php the_post_thumbnail('middle');?></a>
</div>
<div class="text_container">
<a href="https://wordpress.stackexchange.com/questions/242083/<?php the_permalink() ?>"><span><?php echo str_replace(':', ':</span>', get_the_title()); ?></a>
</div>
</div>


<?php
// End loop
endwhile;

// Reset post data
wp_reset_postdata(); 

endif; ?>

Related Posts: