You have an image or link (header, <link... not an <a href) somewhere on their site which leads to your site, on your site you have a php script which catches it and processes it. Here's an example.
On their page, in this case the index page, we might put this tiny 1x1px image.
<img src="http://loggindsite.com/logscript.php?page=index" style="height:1px;width:1px;" />
Then in logscript.php
For this I'm assuming you have a MySQL table called page_logs with three fields (id, page, hitcount). There are a number of ways you culd set this up a little better, for example, if you're planning on logging for multiple sites you would need another field or if you discover you need to hold more information about the individual pages you would need to normalize to two tables (page_logs and pages) but this is good now.
<?php
$page = $_GET['page'];
//replace these with whattever your loggins/db stuff is
$con = mysql_connect($server, $user, $pass) or die('Failed DB Connect");
$db = mysql_select_db($dbname, $con) or die("Failed to select DB");
$sql = "UPDATE page_logs SET hitcount = hitcount+1 WHERE page='".mysql_escape_string($page)."'";
$res = mysql_query($sql, $con) or die("Failed query $sql");
if(mysql_affected_rows($res)) {
//didn't find the page
} else {
//updated it
}