This allowed more efficient development and application of mathematics, with seemingly disparate areas enriching each other, it also became a tool for investigating meta-logical properties of formal theories themselves. E.g. Hilbert showed independence of his geometric axioms by exhibiting models where all but one of them held; Gödel showed consistency of the controversial axiom of choice by building a model with it within the set theory without it; Skolem showed that all first-order theories, including real analysis and even Cantor's transfinite arithmetic, had countable models, etc. Skolem's and Gödel's results underscored the meaning-neutrality of the formal, and ushered in the current dominance of the first order logic, see http://philosophy.stackexchange.com/questions/28253/where-did-g%c3%b6del-write-that-first-order-logic-is-the-true-logic/28255#28255Where did Gödel write that first-order logic is the "true" logic? But they also highlighted the limits to its expressive power, and later undermined the logicist program of Frege and Russell, and the formalist one of Hilbert, see respectively http://philosophy.stackexchange.com/questions/34074/what-is-the-philosophical-ground-for-distinguishing-logic-and-mathematics/34077#34077What is the philosophical ground for distinguishing logic and mathematics? and http://philosophy.stackexchange.com/questions/28154/was-there-a-kantian-influence-on-hilberts-formalist-programme/28157#28157Was there a Kantian influence on Hilbert's formalist programme? Tarski showed how models allow to separate the theretofore conflated (even by Husserl and Carnap) notions of semantic and syntactic inference, which had a profound influence on analytic philosophy of science and language. He also gave a model-theoretic proof of Gödel's celebrated incompleteness theorem. Russell quipped about the import of the nascent model theory in Mysticism and Logic (1917) Ch.4 as follows:
One need only replace "mathematics" with "theoretical science" to recognize the drive behind some of the major trends in the 20th century epistemology of science, see e.g. http://philosophy.stackexchange.com/questions/30033/what-is-the-underdetermination-of-theories-by-evidence-and-how-does-it-square-w/30037#30037What is the underdetermination of theories by evidence, and how does it square with scientific realism?