Why is this?
Many historians avoid “theory” that characterizes the other social sciences. Indeed, there’s a debate about whether history is a “social science” or a “humanity” based in large part around historians avoidance of such theory and laws of history. But one thing historians do engage in is periodization: breaking the endless stream of history into characterizable chunks.
One of the most popular periodizations for the age that comes directly before ours is the Short Twentieth Century (1914-1991). This term was popularized by Eric Hobsbawm, who had also coined the term the Long Nineteenth Century for the period from the French Revolution to the start of World War I.
What I think your teacher is telling you is that communism, and the reaction to communism, defined much of this Short Twentieth Century. This period in the West is, after all, first defined by the fragility of the liberal democracy in the West, especially under the influence of radical anti-communism and a real chance of Communist Parties coming to power in government (through generally in coalition with more moderate Socialist Parties), then as a bipolar world where almost all European and Anglophone nations were aligned either with the the US and NATO or the USSR and the Warsaw Pact.
There are events that I can point that would have happened even without Communism: the emergence of Poland after WWI, which likely would have happened even without the Communist Revolution in Europe and proved very important for the start of World War II. The increasing Western involvement in the Middle East also would have increased even without Soviet backing. Decolonization likely would have still happened, though perhaps more slowly and in a different way.
The thing is, your teacher is right that Communism and Anti-Communism touched almost everything of the things that happened on the world stage, especially after the Second World War, and while it’s harder to say that Communism and Anti-Communism touched every aspect of domestic politics, but they affected a lot of things. In America, anti-Communism became one of the three “legs” of modern American conservatism (along with libertarianism and traditionalism). Democrats, meanwhile, were often afraid of appearing “soft on communism” and by the 1950’s Marxist influence was at a low point. However, earlier, in part due to Soviet inspired and Soviet sponsored organizing, membership in Socialist and Communist parties had exploded (remember, though, that America also had an indigenous socialist tradition even before the Russian Revolution—Eugene V. Debs the Socialist Party candidate earned 6% of the presidential vote despite having no chance of winning the election).
In Europe, though, it’s slightly harder to say how the Socialist parties founded during the Second International (Labour in the UK, Social Democratic Party of Austria, Italian Socialist Party, etc) would have behaved if there were no Soviet Union. These groups generally had Marxist factions before 1917, and continued to have Marxist tendencies after. Again, their context was shifted by the Russian Revolution, but it’s hard for me to say precisely how.
Likewise, it’s hard to say exactly how anti-Communism affected Conservative Parties. Would the Tories have privatized much of the public housing stock in the 80’s without specific anti-Communism, would anti-Socialism have been sufficient? Council (public) Housing had existed in the U.K. back to the 19th century, and really exploded after the Second World War—but it’s hard to connect the growth to the Russian Revolution or Communism, except in that the Russian Revolution was a factor in both the rise of Hitler and the Second World War more broadly, and one of the big justifications for the rise in public housing was to make homes for veterans (as it was after WWI) and much of Britain’s housing stock was destroyed by bombing. Would similar things to this, both the growth of public housing and its later privatization, have happened without the Russian Revolution? Yes, it’s possible. But history is contingent. We have no possibility of rerunning the experiment of time, but we can see that in other European countries, public housing was inspired by Socialist parties but, even in Britain, with some level cooperation from the Conservatives. History is contingent, sometimes elections are decided by tiny things which have huge effects, but it’s almost certain that these same debates about domestic policy would have been carried out with or without the Russian Revolution.
The Russian Revolution did really change the context for the European Left, and therefore Conservative reactions to the Left. One simple thing I can point this out is that there weren’t any things called a “Communist Party” between 1852 and 1917 (again, Marxism had earlier been a tendency within working class parties). Then, there was an explosion of them, looking to the Soviet Union, getting support from it, and more over interacting with the works of Marx very differently. I cover that more in my answer to:
But anytime you hear a Communist or a Communist Party (or an Anti-Communist or Anti-Communist organization) did something, that is a direct effect of the Russian Revolution.
It certainly affected the politics of the West up and down, but I’d be more comfortable saying it affected almost everything than saying it caused almost everything.
In addition to what /u yodatsracist wrote in his very informative post, please take note that even if you leave the communist ideology from the equation, Russian Revolution still remains one of the most influential event in Western (or at least European) history. It directly caused the withdrawal of the Russian forces what heavily influenced the course of the Great War and with the peace treaty after the Great War significantly influencing the political upheaval in Germany it would also influence the Second World War. I don't say Great War outcome would have been vastly different or that there would be no Second World War, but these event would have been different should the Bolshevik Revolution had not occurred.
Anything that passed between 1945 and 1990 West of Rhine is a direct product of Soviet Union forming satellite states, i.e. semi-sovereign states forced to adopt political system and controlled from Moscow (I said 'west of Rhine' because events in the East Germany influenced the politics of West Germany). Anything related to Cold War as we know it is, of course, directly related to the emergence of a strong, aggressive country with vastly different ideology that was Soviet Union (to nitpick, you could say that it was the emergence of Soviet Union that actually changed history, because October Revolution itself could have always failed).
We should also remember that the strong position of the Soviet Union could have helped the Communist Party of China to seize power in the 1950's and the Soviet and Chinese involvement fuelled the Vietnam War that intensified civil rights protests in 1960's in USA. And, of course, the emergence of allegedly communist North Korea (it looks more like patrimonial monarchy, but that's the story for another time) led to events that got the spotlight in recent two years. Soviets were also instrumental in supporting at least several leftist groups in South and Central America. So, summing it up, I'd say that the Russian Revolution not only influenced most if not all important events in the Western history, but also influenced events around the world to the extent making it easily the most important global event in the last century.