Fraught means “accompanied with”. For example:
“Your exam is fraught with errors” is a valid sentence.
“The road ahead is fraught with danger” is also valid.
In this context, emotionally fraught simply means that the conversation is heavily affected by people’s feelings regarding the consumption of meat. Many people love meat and therefore reject plant based alternatives, and might even get upset when someone suggests that they should change their diet. This is not something that would happen if people were talking about smartphones. In that case, the conversation is emotionally neutral - people are able to have a reasonable discussion without getting hijacked by their emotions. The speaker is just making that distinction.
A good way of putting this, much more common in everyday English, is to say something is a “touchy subject”. If you say that something is a touchy subject, you mean that it should be talked about in a very sensitive way, because it might upset people.
Some examples:
“Don’t mention the recent election, it’s a touchy subject”
“Politics is a touchy subject”
“When you see her, don’t mention her recent exam, it’s a touchy subject”.
Hope that helps!