First, confirm you’re using the correct (i.e. latest) versions of all plugins — fairly often I’ve heard from people using either an old version of CF7 or an old version of my plugin; there was a major change to CF7 a while ago that required I update my plugin accordingly so you need to use the newer versions of both.
After that, open the browser developer tools (usually F12) and go to the sources/console tab to see if there are any Javascript errors both on page load and when you submit the form. Then, check the Network tab when you submit the form — you should see a new POST entry, and you can see the response. If it looks weird (i.e. not a JSON success response), then something server-side is broken.
If it’s something server side, you can try enabling WP_DEBUG and WP_DEBUG_LOG via your wp-config.php
file and check the wp-contents/debug.log
file to see more info about what broke.