The response code says 500 (Internal Server Error). That means your jQuery code is probably OK, because it seems to be sending the request to the PHP backend. However, the 500 status code means the PHP backend is crashing.
What you can do is enable Display Errors through the control panel (see the link KangJL shared), then open the web page, open the Network tab in the developer tools of your browser and try submitting the form again. In the Network tab, you should be able to see the request to the PHP code, and see the content being returned. You should see a PHP error message there, which you can then try to tackle.