The short version as i understand it is that Reagan *did* have an important roll in bringing the Cold War to an end, but it was more because of his willingness to talk with the Soviets and give them a way out and less the saber rattling. That the biggest problems the Soviets had were self-created through the belief that they needed more control over their people and over the ideological discourse of the nation. By clamping down on their populace, they made western culture look more appealing.
This is certainly part of it. Although if any leader other than Gorbachev had been in charge, it might not have worked.
The USSR was always going to be suspicious of US intentions and somewhat hostile. Political paranoia has a long history in Russia...historically speaking, Russia wasn't so much a nation as a transit route for up and coming imperial powers, either from the east (the Golden Horde) or the west (Teutonic Knights, Baltic princelings). Often these two forces acted in tandem, destroying what little political authority native princes and lords had, at the very least up until the time of Tsar Ivan IV (the "Terrible").
As such, political authority in Russia was always sharply centralized, and often concerned about being "encircled" by foreign threats or being subverted from within. Before the KGB, the Tsars ran a highly effective and dangerous secret police force, one which had infiltrated many terrorist networks in the country and were responsible for elaborate forgeries like the Protocols of the Elders of Zion, a favourite conspiracy screed of anti-Semitic groups even to this day.
Bolshevism merely updated many of these traditional approaches and trends within Russian political culture. That almost immediately after the end of WWI, while the civil war still raged, that the "capitalist powers" of the UK, America and France intervened on the behalf of the "Whites", many of whom were almost proto-fascist in their worldview, was not quickly forgotten, even if their contributions were overstated. And that in the decade before WWII there was an unofficial policy of attempting to use the fascist states of Japan (East) and Germany (West) to contain international Communism certainly awakened old fears concerning encirclement, which eventually became official US policy after WWII, with the concept of containment.
Russia, as a state, was paranoid. Stalin in particular, literally so. Recent theories suggest, with some backing, that he had a brain disorder that made him a clinical paranoid, he saw enemies and conspiracies everywhere. But even without that, the political culture of the Soviet Union, on the bedrock of Tsarist suspicion, worsened by a brutal civil war which led to the formation of an unaccountable secret police force, dragged to the brink of starvation in the 1930s, brutally invaded by a former ally in the 40s...you had a country inclined to believe the worst of others and mistrust their motives even at the best of times.
The ideology of Communism did not help, of course. Communism was meant to be an advance on capitalism, a resolution of its inherent contradictions into a higher political state of being. And while the Soviet Union did manage some astounding economic growth, which likely would not have happened as quickly without Communism, it also capped its potential.
This paranoia also led to the establishment of a vast "counterintelligence state" and massive military buildups, which of course diverted funds which could have been used elsewhere. Some people credit Reagan with forcing the Soviets to spend outside of their means, triggering an economic crisis and collapsing the USSR. This is partially true, but not the whole story.
Firstly, despite certain economic weaknesses, the Soviets still had deep pockets, namely in the form of massive natural energy resources that they could sell for foreign currency or use to underwrite the economies of their Warsaw Pact allies. However, the Saudis boosted production massively in the 80s, causing the price of oil to crash. They knew what they were doing, and it worked - the Saudi gambit hurt them a little, of course, but they'd been prepared for it - the USSR had not. Suddenly, the economy was in a massive contraction, while military costs continued to rise due to Reagan's buildup and the war in Afghanistan.
Gorbachev rightly concluded that this could only be combated by making peace with the West, and abandoning the Warsaw Pact states to their own fate. In previous decades, native armies had been used to put down uprisings in most states, because it was known what the Soviet Union would do if it had to use its own forces (Hungary 1956, all over again). Local communist parties preferred to use their own forces to save their own necks. Army officers further up in the chain of command also knew the score. But with the threat of Soviet intervention gone....they didn't know what to do. Should they carry on as before? Didn't work out well, in Romania certainly. Hungary flipped sides (due to West Germans calling in their loans) and the border was opened with Austria...given free movement between states like East Germany and the Hungarian Republic, suddenly, many people in the Warsaw Pact could freely move to the West. Unless states moved quickly to liberalise their economies and their politics, they'd suffer a mass exodus that they couldn't stop.
Hardline elements within the Kremlin, KGB and military saw this as a disaster, and used it as a pretext to overthrow Gorby in a coup. Only, well, we all know how that worked out...