There is an exchange of a cookie which carries a unique, pseudo-random session ID. Session data for each different user/session is stored with a file name based on that same ID.
So, all you need to do is initialize a session with session_start(), and it automates everything concerning looking for that cookie, setting it for a new session, etc.
In your email script:
<?php
session_start(); // must come before *any* type of output.
// save data in session:
$_SESSION['post'] = $_POST; // simple solution, you could save specific fields if you want
// send your email, then redirect to success page, etc.
?>
success.php:
<?php
session_start();
echo "Thanks for your email, " . htmlentities($_SESSION['post']['name']) . ".";
?>