TLS is a protocol used to encrypt traffic, for example web browsing (HTTPS). To do this you need to do a handshake where you establish the encryption keys (which are used to encrypt the traffic). But there is nothing stopping someone sitting in the middle of this handshake from just doing the handshake with each part and then reading the traffic, which defeats the entire point of encryption. You need assurance that you are actually talking with who you want to be talking to.
This is where certificates come in. Certificates contain the domain name, the server’s public key, and are signed by a certificate authority that vouches for the authenticity of the certificate. Your OS or browser has a store of CA certificates that are trusted. These certificates contain the public key of the CA. This public key is then used to check that the certificate the server presents is valid. To get a CA to sign a certificate you have to prove you have control over the domain name, this can be done in a few different ways and is usually automated.
The problem in the post is that someone has found the private key, which is used to do the signing, of one of the authorities. This means they could create a certificate for any domain name that they want to, and browsers/other software would accept as valid

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