<?php wp_footer(); ?>
WordPress features an incredible hook API that allows developers to put their code in almost any part of the site without modifying anything, but to do this, there must be called special functions like wp_head() or wp_footer().
I don't understand why, but there are some theme developers that remove wp_footer(). This not only breaks WP Carousel, but any other plugin that uses the hook wp_footer to put important code of the plugin.
The easiest way to fix the problem is editing footer.php file of your theme and add near </body>
<?php wp_footer(); ?>.
With that WP Carousel's code will be loaded.