Bumble's answer, already selected as the answer, outlines what Dummett knew, especially being an expert in the works of Frege, the widely acknowledged father of the analytical tradition, and a key contributor to contemporary discourse surrounding realism and anti-realism. Understanding the gulf between various positions of realism and anti-realism is important because it highlights major philosophical divisions in mathematics (neo-Platonism on one end and perhaps Brouwer's intuitionism or Pierce's empiricism in the other extreme) and science (perhaps a spectrum from scientific realism to Duhem's instrumentalism). And all of these divisions and disputes hinge upon, literally a turn of phrase. It is indisputable that language is indispensable to philosophy as a social activity and a means of creating personal meaning. Therefore, the philosophy that delves into the nature of language to a large extent determines the nature of other philosophies, because meaning itself is determined largely by the public artifact of language, a key concept Ludwig Wittgenstein explored. If you doubt it, provide of an example of philosophy where words aren't used. They're few and far between.
Contemporary philosophy of language owes a debt to both Pierce and de Saussure, and their contributions to understanding language as a social process, the use of signs in society and how meaning is constructed is undeniable. But much of their work was embraced by philosophers in Europe who had an interest in discussing philosophy as tool to understanding man in the context of society and politics. Continental philosophy is not so much a method as a devotion to contextualizing ideas within a social context. One need only think of Habermas, Foucault, Sartre, Camus, Derrida, Gramsci, and Arendt to see what analytical philosophy is not, and yet these philosophers weren't ignorant about language use. Deconstruction is very much a philosophy that is built on a view of language. Phenomenology repurposes language to explicate conscious existence. Philosophical anthropology recognizes how language bridges the gulf between human beings. But none of these really ask the question. What is going on with the syntactic-semantic divide? How do words bear upon truth? Do they merely bear the truth in propositions, or do they fundamentally determine it in some way? This, then is where analytic philosophy differs from the practices of the more European tribe of thinkers.
Russell should be recognized as transformative not only because he recognized the genius of Frege and Wittgenstein, or because he put forth a logical edifice with A. N. Whitehead that attempted to create the logical equivalence to Euclid's Elements, but because through his interest in paradox in language, he explored how logic relies on language and grew the symbolization of language that Frege had started with his simple formal systems. Russell's work in sets and particularly his type theory created a model for building language systems that is used for mathematical logic, computability theory, and modern computer languages today. The intersection of mathematics and logic, central to the logicist project, is fundamentally one of language. Kleene, Church, Turing, and Goedel were the sorts of men who combined math and logic in the same way that Mauchly and Eckert built physical systems that automated math and logic. The modern computer runs on an ALU, an arithmetic-logic unit. Thus, analytic philosophers have not only explored syntax, they have built machines that have elevated syntax to a medium of art, and if the computer scientists Newell and Simon are to be believed, the essence of intelligence itself. Computers are nothing more than the embodiment of the language of mathematical logic. This is about as far removed from the concerns of the Continental philosophers as one can get.
While early Wittgenstein was part of this movement with his Tractatus Logico-Philosophicus, late Wittgenstein went in a different direction, and began, as the Continental philosophers do, to reflect on language as a social enterprise. Three major contributions to philosophy are his notions of family-resemblance of meaning, the private-language argument, and the language-game. Along with Austin, Strawson, and eventually Searle, a philosophy of language has emerged as a fully fledged field of inquiry that makes use of the insights of literature, anthropology, linguistics, and cognitive science to explore just what language is. And unlike the artificial languages of mathematical logic whose primary preoccupation is truth-valued semantics, that is where meaning is largely determined by truth itself, modern linguistics and the philosophy of language that surrounds it has a variety of theories of semantics, some orthodox, and others not. Thinkers like Chomsky, Lakoff, and Jackendoff have teased out a view of how the brain is related to meaning, truth, and grammar, and have a mountain of scientific research to speak from. Chomsky rekindled cognitivist perspective when the Academy was rife with behaviorists. Jackendoff has a conceptual model that interrelates syntax, semantics, phonology, and other aspects of language supported by cognitive neuroscience. Lakoff has philosophical speculations that attempt to reorient semantics beyond truth-conditional sentences and in bodily experience. In fact, the philosophy of mind has never been more preoccupied with language use.
Analytic philosophy, in all its flavors, then presumes mathematical logic and scientific research as central to its interests and methods, and both of these endeavors rely heavily on an understanding of language. Logical positivists hit a wall with their observation sentences and underdetermination of theory, and the subsequent logical developments like Quine's attack on the analytic-synthetic dichotomy as well as alternative foundations of mathematics such as category theory have thoroughly defeated logicism. Post Kuhn, science and mathematics are engaged in a scramble to re-orient themselves in the continuing debate between realists and anti-realists. And all of these debates center around one's view on language. And Dummett's claim to fame is furthering this debate.
Moritz Schlitz says "We see that meaning and truth are linked together by the process of verification; but the first is found by mere reflection about possible circumstances in the world, while the second is decided by really discovering the existence or non-existence of those circumstances" in "The Future of Philosophy". Philosophy's purpose is to referee that process, and the tool to conduct it is language. Philosophy as an enterprise cannot be reduced to philosophy of language, but no philosophy that ignores it can be complete or reliable. That's all Dummett is claiming.