Upload all the .php files except config.php, install.php, and upgrade-3.2.3.php. Don't forget to upload the .php files in the sources and languages directories.
Go to upgrade-4.0.x.php in your browser and click the "Upgrade" button.
Everything should be working now. If you didn't make any major changes to the templates, it is recommended that you upload the new templates directory to take advantage of some new features such as up/down/neutral arrows and alternating colors. Your old templates will still work fine, though, you only need to upload the new admin.html, lostpw_form.html, lostpw_finish.html, and gateway.html.
Install the script like normal. Upload upgrade-3.2.3.php. Go to upgrade-3.2.3.php in your browser and follow the instructions.
If you want your old members to keep the same HTML code in their website, replace your old topsites.cgi with the new one included in this zip. Just change $topsites_php_url to the url to the directory you're installing Aardvark Topsites PHP in.