.. and our surveys said... EH UHHH ...
Your most major problem is your form has no action...
<form action="" method="post">
.. should be:
<form action="<?php echo $_SERVER['PHP_SELF']; ?>" method="post">
Also, that script appears to have been written for GLOBALS on, which is just plain bad.
So for the moment, try changing your form action to mine above. That should fix it.
Then you can start doing simple PHP and learning why that script is just wrong 😉