Yeah, there are a number of ways of telling someone how to shut up in French, (and Quebec does it differently than France). In France, this is one of the more aggressive / assertive / rude ways. But, the way he said it was a milder version, so it's more a "stfu" rather than "shut the fuck up".
There's "tais toi" which is basically like "be quiet". What you might say to kids being annoying. There's "ferme ta bouche", which is literally "shut your mouth". Or, "ferme la", which is basically "shut it". He basically picked the rudest form, but said it in the mildest way.
"Ferme ta bouche" (shut your mouth) or "ferme la" (shut it) are not mild, but not as strong as "ferme ta gueule" or "ta gueule". I don't know if there's a real difference between "ferme ta gueule" or "ta gueule", it probably depends more on how it's said than the words. Once you start using "gueule" it's a pretty strong form.
Really the two words "ta gueule" is shorthand for effectively saying "you have the mouth of an animal (a predator?) not a human, and you should shut it".
The closest English equivalent is "shut your hole".
Some debates don't even need to happen anymore. No the earth isn't flat. No trans people aren't monsters trying to kill Jesus. No vaccines don't turn you gay. No the world wasn't created 6000 years ago.
We've legitimized too many morons in the world. Not everything is a debate, some things are just facts that need to be thought to the ignorant.