You probably want to use cURL.
http://www.php.net/manual/en/function.curl-exec.php
The $xml variable seems to generate the URL you are going to query (after replacing SERVER/PATH with the proper usps server/location to query). Pass that in as the URL as seen in the cURL example on the linked page.
The XML you get back from curl_exec should look like the response XML you included. From there, PHP has a xml reader class that you can use to extract the information you want (or if you just wanted to grab the 'rate' then regex might be a little more lightweight).
Edit: In case you miss it in that documentation page, CURLOPT_RETURNTRANSFER flag needs to be set for curl-exec to return the contents rather than just "SUCCESS".