You need to tell the PHP to echo the div.
if (! is_front_page()) :
echo '<div class="button-kontakt"> <a href="https://wordpress.stackexchange.com/kontakt"><p>ANFRAGE</p></a></div>';
endif;
Unless you just want it on the top of the page, you need to add that code to a template file, not the functions file. If it is something that appears at the bottom of every page except the homepage, putting it in footer.php may be applicable.
Also, have you set the page you don’t want to see it on as the front page via the option in the Appearance->Customise menu?