Let's take Max Horkheimer's definition of critical theory. He wrote that a theory is
critical insofar as it seeks "to liberate human beings from the circumstances that enslave them."
The impulse to this idea came to Horkheimer in WWII as he experienced that fascism was already encoded in the notion of enlightenment and reason. Thus he and Adorno wrote in The Dialectic of Enlightenment that "What man wants from nature is to dominate it and other men."
So you see that critical theory is all about liberation and enslavement and domination.
But where does critical theory come from?
Oddly, it has nothing to do with leftism. It goes back to Kant and his Critique of Pure Reason.
Accepting Hume's argument that correlation isn't causation, Kant argued that there could not be absolute knowledge because we could not know "things-in-themselves" but only appearances. So all human knowledge is a story, a "narrative" about things observed in the world, that tries to make a connection between appearances and the unknowable "things-in-themselves." Notice that if we can't know "things-in-themselves" it means that we never know whether our last scientific theory is really the last word on reality.
Then Fichte argued that "All our thought is founded on our impulses," a creative act of imagination, the first stirring of the idea of the unconscious mind.
If thinking is not the mechanical application of reason and logic, but impulsive narrative, it stands to reason that powerful people have the power to impose their impulses and "narratives" on the less powerful. Thus, it makes sense that the experience of marginalized peoples will be suppressed by the narrative of the ruling class. Thus Critical Race Theory.
Critical Race Theory is how most of us encounter critical theory, and CRT takes critical theory to the max. Here are the highlights of Wikipedia's breakdown:
Critique of liberalism e.g., Enlightenment, rationalism, legal equality.
Storytelling, "naming one's own reality" "The use of narrative (storytelling) to illuminate and explore lived experiences of racial oppression."
Revisionist interpretations e.g., that civil rights laws were passed "in order to improve the image of the United States" abroad.
Intersectional theory Intersection of "race, sex, class, national origin, and sexual orientation, and how their combination (i.e., their intersections) plays out in various settings."
Empathetic fallacy "Empathy is not enough to change racism" because most people live in a bubble.
Non-white cultural nationalism argument for racial "separation and reparations" for oppressed peoples.
What is interesting is how left-wing thinkers and activists take the notion of critical theory, that all knowledge is a narrative of power, and demand that we accept their critical theory as Holy Writ. Or else. Narrative of power, anyone?
Anyway, Critical Race Theory is based on the notion that only the powerful have a voice, but that the non-powerful have experience that is valid for them, and who is to say that their experiences are any less valid that the narrative of the powerful.
But what if the powerful people are right about a few things? For instance what if capitalism creates prosperity for the marginalized worker while socialism creates killing fields? What if an education without a lefty in sight creates capable people while graduates in critical race theory turn out to be mind-numbed robots?
If we accept the ideas of Kant and Fichte then it is not hard to see how they license the impulsive ideas of relativity and quantum mechanics, that completely liberated us from enslavement to the bonds of the mechanical world of Newton.
Critical theory has been an astonishing success in science. But it is much easier to determine whether an impulsive idea in science works, and much harder to do that with ideas about human culture and politics.
Climate Change illustrates the operation of critical theory. How would we know that the scientific theory of climate change is correct? We won't, not for decades or centuries, not until critical discussions among scientists have understood climate much better than now. But meanwhile the rulers and the experts want us to act and they do not permit a critique of climate theory from people without power. How do we know if the critical race theorists are right? We don't, because you are not allowed a "revisionist interpretation" on race, so we either acquiesce to the dominatory ideas of the rulers or we oppose them, even unto death.
Lefty Jürgen Habermas tried to get beyond the opposition of Enlightenment vs. a liberated world free of hegemony and domination. He developed the idea that we are not just mechanical wind-up toys, cogs in a rational machine or helpless victims of the powerful, but communicators. So his Theory of Communicative Action proposes that the systems world of Enlightenment and instrumental reason should be balanced by a life-world of communication and negotiation.
Notice that the "life-world" of Habermas understands the critical race theory's notion of "lived experience." But in his world the differences in lived experiences are there to be communicated and negotiated not fought out in the streets with mostly peaceful protest or imposed in the classroom by government teachers.
The notions of critical theory have interesting ramifications in the work of Thomas Kuhn and his Structure of Scientific Revolutions. How does a new scientific theory gain a foothold? When the old generation of scientists dies off.
Strictly speaking, critical theory ought to be the standard of the truly marginalized in their battle against The Man. Yet in our day critical theory is the weapon of choice for our ruling class and its supporters in education, media, and politics against the ordinary middle class.