President Reagan's no-nonsense attitude toward the Soviets scared them for the first time. Before that, they had had American presidents wrapped around their little finger - remember Jimmy Carter smooching with Brezhnev? But when Reagan began making jokes about starting the bombing in five minutes, and calling the Soviets an Evil Empire and the focus of evil in the modern world, that scared the living daylights out of them. KGB files prove it.
In 1979, President Carter had indeed given Soviet leader Brezhnev a kiss, in a clumsy attempt at what Carter thought was a Slavic custom. But massive amounts of information from the KGB were made available after the fall of the Soviet Union, which detailed Soviet understandings of Reagan's seriousness.
With the Reagan defense buildup, we showed that could maintain a world-class defense and a first-class economy. And we showed that the Soviets could not. They crumbled trying to keep up. They couldn't feed their people. Here was a country that could build state-of-the-art tanks but could not build a washing machine to last five days - or deliver it to the buyer earlier than ten years from the purchase date. That's what you get with a command economy.
The basis of America's winning strategy in the Cold War was economic, not military. Buy engaging the Soviets in an arms race, we forced them to overextend themselves financially, and it was their budget which eventually did them in.
As soon as people in the Soviet Union got the freedom to say what they really thought, many of them began to say very openly that their government was, in fact, an Evil Empire. (What else do you say about a system that murdered at least 40 million of its own people?) And often they quoted Reagan's actual words, because, of course, Reagan's "Evil Empire" speech had been widely publicized in the Soviet Union as evidence of the nefarious designs of American internationalists.
There is a great deal of irony in the Cold War. One bit of irony is that Reagan was extremely popular in Russia after the fall of the communist totalitarian government. That irony is compounded by the fact that Reagan had become so well-known, and so well-liked, among the Russians because that communist government had widely circulated Reagan's speeches, in the hopes that it would build Soviet resolve against American. The publicizing of Reagan's words had the opposite of the desired effect: by the time communism fell, Russians were familiar with, and enthusiastic about, Reagan's speeches.