The topic is somewhat vast due to the vagueness of "language" and "language use", so the claim can be construed to implicate most of the analytic philosophy. As Dummet writes in The Logical Basis of Metaphysics
"It has until recently been a basic tenet of analytical philosophy, in its various manifestations, that the philosophy of thought can be approached only through the philosophy of language. That is to say, there can be no account of what thought is, independently of its means of expression..."
Before that some specific suggestions as to what (broadly linguistic) means thinking "requires" were made by Aristotle (phantasma), Kant (schemata) and Peirce (diagrams). Empirical studies over the last century made any such specialized "vehicle of thought" highly implausible. Brandom's is not that kind of claim, and I will focus on his. He allows that rational thinking may employ non-linguistic means, but for it to develop in the first place requires participation in what Wittgenstein called language games. More precisely, a particular type of language games, what Sellars called "giving and asking for reasons". Here is from Reason in Philosophy:
"The game of giving and asking for reasons is not just one game among others one can play with language. It is the game in virtue of the playing of which what one has qualifies as language (or thought) at all. I am here disagreeing with Wittgenstein, when he claims that language has no downtown. On my view, it does, and that downtown (the region around which all the rest of discourse is arrayed as dependent suburbs) is the practices of giving and asking for reasons."
In other words, the rationality is "parasitic", as Brandom puts it elsewhere, on "giving and asking for reasons". Let me try to flash out Brandom's "syllogism".
1) Rational thought is about imparting and understanding meaning.
2) Understanding meaning requires making inferences, meanings are inferential roles.
3) Learning inference requires norm-governed (hence communal) linguistic practice, "giving and asking for reasons".
Ergo. Let's say that 1) is vague enough to be relatively uncontroversial. 2) is the thesis of semantic inferentialism, which Brandom credits to Hegel (ultimately), although his own version is more reminiscent of late Wittgenstein and Sellars. What goes for inferentialism is that its traditional alternative, representationalism, ran into some intractable difficulties in semantics and epistemology, see What is the difference between expressivism and representationalism in modern philosophy of language? The best known are perhaps Wittgenstein's rule-following regress, especially as laid out by Kripke in Wittgenstein on Rules and Private Language, and Sellars's argument against the Myth of the Given touched on in How is the conflict between created by reason and external aspects of knowledge resolved? Brandom has a study guide for Sellars's Empiricism and Philosophy of Mind, where it was originally developed (some of its ideas were anticipated by Hegel, Peirce and Adorno, among others). Quine's indeterminacy of translation and Davidson's indeterminacy of interpretation also belong to this circle of arguments. Verheggen in How Social Must Language Be? gives an insightful commentary on Davidson's arguments in comparison to Wittgenstein's.
3) is a major theme of Wittgenstein's Philosophical Investigations, although, unlike Sellars, he talks of language as a "motley", and does not single out giving and asking for reasons. Still, Wittgenstein's private language argument suggests (vastly oversimplifying) that signposts can only point if there is a custom of following them, and hence that normativity requires communal practice. A language may be privately used but not privately developed and established, that "requires something independent". The argument is discussed under Did Wittgenstein consider the possibility of a private language with public content?
All three arguments are very complex and intricate, we can do them only so much justice in SE format. But I want to point out a peculiar feature. They are not so much arguments for Brandom's semantic pragmatism as against the existing alternatives. This is not accidental, and reflects a different approach to arguing a position, first championed by Peirce, see Do all epistemologies suffer from the "regress of justifications" problem? It is unlike that of classical rationalists, Descartes, Kant, Fichte or Husserl, who tried establishing positive claims from imaginary "ground zero". If Brandom manages to recover traditional semantics in its confines (established disciplines with well conceptualized domains), and give an attractive account of new concept formation and acquisition, which evades realist semantics and epistemology, his position will have to be accepted without any (traditionally) positive argument for it, as a better alternative. He names as much as his task since Making It Explicit.