Being aware of and actively attentive to important facts and issues is not bad. Using language that implies that you, but not other people, are aware of and actively attentive to important facts and issues is bad. And using language like "woke" which implies that other people are asleep or they would agree with you, is bad.

It's similar to using words like "sheeple" to describe people who you disagree with. As you said, it's impossible to take you seriously when you use terms like that. That's what the outraged conservative commentators are latching on to. They're telling their listeners, "They think they are woke and you are asleep". Things are tribal. It's a powerful message.