Hi Fm,
as you describe it, you obviously followed all the steps correctly to get your first push notification:
1. Generate your own VAPID keys as described in readme.md and 
- enter the public key in PNServiceWorker.js
- enter both keys and email in PNTestServer.php in the following place
    // set the VAPID key
    $oVapid = new PNVapid (
            "mailto: 
[email protected]",
            "the-generated-public-key",
            "the-generated-private-key"
    );
2. Open the PNTestClient.html page
-> Click the [Subscribe PUSH Notifications] button
-> To check, reload the page again ([F5])
-> "PUSH Notifications are subscribed" should now be displayed
3. Open the PNTestServer.php page
-> the first notification should appear on your PC next to the information displayed on the page
From here it is difficult for me to narrow down a source of error without further information if the notification is not displayed.