The flow would be this:
1) Browse to the form you want php to fill out, do a view source and grab all the form's input field names and create a map that tells you what information to put into each field.
2) Write a php script which
a) Opens a curl connection to the desired web page ([man]curl_Init[/man]).
b) Sets the appropriate options on your curl session ([man]curl_setopt[/man]) The options you'll definately want to pay attention to are CURLOPT_NOBODY, CURLOPT_POST and CURLOPT_POSTFIELDS. However go over the complete list and see if any others should be used as well, i.e. setting CURLOPT_TIMEOUT is usually a good idea. Also be sure to read the user comments as those are what really explains the parameters to you.
c) execute the curl_session ([man]curl_exec[/man]
That's really all there is too it, don't let curl scare you it's very simple to use and has a rather small learning curve.
NOTE: If the form you want to fill out requires that you log in then you'll probably want to submit the log in form first and use the CURLOPT_COOKIEJAR parameter. Then you can submit the form you really want to submit using the CURLOPT_COOKIEFILE parameter and pointing to the file created by the CURLOPT_COOKIEJAR file from the first session.