How exactly can an infected Samsung TV record private conversations in a room?

The TV is not "hacked" the Samsung User Manual Privacy Policy says in the Fine Print: "capture voice commands and associated texts so that we can provide you with Voice Recognition features and evaluate and improve the features" - it's a Diagnostic Test Mode.

The TV may appear to be Off but there's a blue light on the back that is On.

To disable: Head to your television’s settings menu, choose “Smart Features” and then move down to the “Voice Recognition” feature and switch it off.

A partial answer:

Which models are safe

All models which don't have any microphones built-in.

How this bug exactly works

The how is easy. If your TV has a microphone (for example your remote might have one to enable you to control the TV via voice recognition), an attacker can turn on the microphone, record everything that it picks up and send it to a server on the internet where it can be picked up by an attacker. This is the same kind of attack that can be done on smart phones, except that there an attacker can also turn on the camera without your knowledge. This is why some people put a tape over the front camera of their smartphone and the builtin webcams of their laptops.

While André's suggestion not to connect the TV to a network might not be practicable, it's really the only way to be sure that your TV doesn't spy on you (the microphone recording is the biggest problem, but the TV can also spy on you by recording what you watch and when you watch it). I'm pretty sure netflix does this as a normal part of their business.

Another solution, if you're not afraid of operating on your TV set, would be to physically disconnect the microphone.