After redirection to the homepage Acymailing displays a message in your system message container. I checked and you don't have the Joomla system message container in your site. Please, add it to your template or make an override for your index.php and add this some where on top of your page:
Once done, check again. You should see a confirmation or an error message. In case you get a confirmation and the subscription of the user is not modified, please let us know.