The browser checks for validity by ensuring the website certificate is signed by a certificate that is shipped with the browser. These "root certificates" are usually owned by Certificate Authorities, such as Verisign or any other number of CAs. CAs ought to verify that the entity creating a new certificate is who they claim to be (the website owner) before signing a certificate. This way, you trust Verisign to tell you that you can trust the target website.
What Kazakhstan has done is create their own root certificate and asked people who live there to install it in their browsers. They are also intercepting any connection to facebook.com and giving your browser a Kazakhstan-created certificate, which is then verified against the Kazakhstan-owned root certificate. Since it will pass this check, the browser shows a happy green banner, even though the certificate is owned by Kazakhstan and not facebook.com. In other words, the data people in Kazakhstan send to facebook.com is now being intercepted and decrypted by Kazakhstan before being forwarded to facebook.com. Facebook is the example used in the linked bug, they can perform this with any other website, too.