I'm not a historian of philosophy, however, I'll have at it. Why was there a linguistic turn which led Philosophers to become fixated on language, particularly in Anglo-American universities? Because disagreements in meaning ultimately unravel into discussions of language. Starting in the late 1800s, linguistics, psychology, and physics created a new metaphysics which allowed a departure from the long-held philosophical mooring of transcendental and absolutist thinking. Part of this new metaphysics allowed linguistics to serve as a platform for the philosophy of language which had a profound impact on all philosophy in the Analytic tradition.
Historical analysis is a tricky business, be we might be to outline some major events that shaped why linguistics became so important to philosophy. Claims that physics and psychology "take away turf" from philosophy is beside the point. It is more relevant that they add to philosophy by way of making decisive some claims over another. In the case of psychology kindled by Wundt and Freud, and others, a certain sort of dispute became settled philosophically with Bolzano's antipsychologism being evolved and championed by Frege and Husserl. One might see this as an apex culmination of German Idealism which has among its Leitmotiven, transcendentalism and absolutism.
For many centuries, philosophy, fueled almost exclusively by the Scholastics, was very much a spiritual, emotional, and romantic pursuit. Philosophy was personal, because it was salvation. Preoccupation with a grand metaphysical framework to bring all intellectual activity under the umbrella of God was and for many is a powerful emotional drive. This began sustaining attacks under the mathematization of physics and the development of science in its modern form starting with Bruno, Galileo, Bacon, and Newton. It's easy to cite Copernic, Hume, and Locke as building up to Darwin as perhaps the apex of assaults on the Great Chain of Being which led many intellectuals to accept a very different cosmogeny, one that was impersonal and mechanical and subject to mathematical analysis.
You see, when theology lost its monopoly over the Weltanschauungen of great thinkers, and the scientific literati became more influential in the great universities of Europe, it seems reasonable that it placed certain stresses on those long traditions of transcendence and absolutism. Eminent thinkers could no longer point to the Bible as a basis for their first principles and instead had to have naturalistic foundations of metaphysics. By the end of the 1800s, with Germany, England, and the US being the leaders in philosophical discourse, that foundation became psychology and physics, the former because it is the science of the mind, and the latter the science of the body corporeal. As you are likely aware, from the Ancient Greeks and their naive realism to Cartesian mind-body duality, metaphysics constantly struggles with ontological questions regarding the concrete and the abstract. Enter psychologists, physicists, and linguists.
Just as psychology was born from the hard science of Wundt and the pseudoscience of Freud, so too were a number of revolutions happening in physics and linguistics. By the early 1900's relativity and the Copenhagen interpretation had transformed Newton's conception of absolute space and time into something unrecognizable. Psychology and neurology had firmly demonstrated under naturalism how central the brain is to the mind and language replacing the Cartesian pineal gland and God as pat answers; Broca and Pavlov are practically household names to the well-educated because they mechanized the brain and showed its correlation (not cause, mind you) to language and behavior. By the time the first quarter of the century had come, Russell and early Wittgenstein had taken Frege's ideas and elevated them with the development of logical atomism which logicians eagerly devoured. By the time the 30's came around and Erkentinnis was being published by the logical positivists, both transcendence and absolutism were nearly dead in the mainstream of the Anglo-American movement known as the analytic philosophy. The followers of Husserl, Hegel, and the other contemporary Continental traditions, of course, kept to their side of the schism, but even these schools were influenced by Saussere and later thinkers like Sartre and Derrida rebelled against the transcendentalism and absolutism of the 18th and 19th centuries. Western philosophy was irrevocably transformed with the birth of so many of the sciences that came to fruition in the Great Industrialization of German, the UK, and the US.
What was the nature of that transformation? I believe it was arguable to say that when the human cosmogeny was rewritten, instead of thinkers striving for union under a theologic motivation, a shift came to explain the diversity. Comparative philology was instrumental in bringing about a new understanding regarding the mechanistic nature of the mind at a time before fMRIs and DNA. That is to say, that language, in a very immediate way, became a window into the operation of the mind liberated from the tyranny of Cartesian thinking. The great German Idealists gave way to the more pragmatic thinkers of the Anglo-American and Logical Empiricists who were very much scientists and logicians in their preoccupation, while the strong emphasis on the self-reflection of phenomenologists and idealists became the Frankfurt School and some of the more contemporary forms of Marxism embraced by structuralists and deconstruction. In either case, transcendentalism and absolutism gave way to relativism and naturalism. Faith and passion became less important than skepticism and empiricism.
So in the Analytic school, philosophers, whose method is essentially linguistic and rational, turned an eye on themselves and their own discourse, perhaps culminating with the Carnapian rejection of all metaphysics outside of that which was subject to operationalization and syntactic analysis. The philosophy of language exploded, not because physics and psychology stole turf (whatever that means), but rather because the rejection of the transcendental, absolutist metaphysics that had held philosophers since Aristotle was co-opted by Aquinas finally gave way to the fracas that was the Pre-Socratics. Thinkers were liberated from a thousand years of Church orthodoxy and influence to see the secular world as it is, not as religious orders want it to be. And what is the best way to settle arguments in the fracas? For scientists, it's devising and running an experiment. For philosophers, it's to use words. And only a fool would engage in debates and logic ignorant of the science and philosophy of language. Syntax, semantics, implicature, intentionality, speech acts, propositions, philosophical modality, semiotics, and logic are the mechanics of the philosophical method and (for the analytical philosopher), doing philosophy without a general awareness of these fundamentals is like doing science without mathematics.
Why was there a linguistic turn which led the anti-foundationalist Rorty to title his book as such? Because disagreements in meaning ultimately unravel into discussions of language. And why did it happen, because linguistics, psychology, and physics created a new metaphysics which allowed a departure from the long-held philosophical mooring of transcendental and absolutist thinking.