Your problem is that PHP will not execute in a *.html file. It will
only execute in a *.php file. Try changing the file extension and
giving that a go.
If that doesn't do it then your other problem may be that the way you
have it set up now requires all of the files to be in the same folder.
You get the error cause PHP can't find any of the files. So make sure
that in one folder you have "verify.php" "recaptchalib.php"
"thecodefortheform.php". Make sure that you download
"recaptchalib.php" at the below link and put it all in the same folder
as well.
http://code.google.com/p/recaptcha/downloads/list?q=label:phplib-Latest
Zack