The Cold War was not a war like World War II that had a very clear beginning. Instead, it was a conflict that built up gradually over time. This is why it is very hard to identify how it started. If I had to say how it started, I might say that it began because the Soviet Union failed to keep the promise that Stalin made at Yalta to let Poland hold free elections. Alternatively, I could argue that it began because WWII ended.
The basis for the Cold War was laid when the Western Allies were strongly opposed to the Bolshevik Revolution in 1917. The Allies even sent troops from WWI to help the White Russians fight the Reds in the Russian Civil War. Clearly, the Western democracies did not trust communism, which helped make the communists less likely to trust the West.
The distrust between the two was exacerbated before and during WWII. Before the war, many people in France and England thought that their countries should side with Nazi Germany against the communists. During the war, the Soviets felt that the Western Allies were content to let the Red Army fight and die against the Germans instead of opening a second front in Europe and taking pressure off the Soviet Union. To the Soviets, this looked like the West was trying to weaken them.
As WWII ended, the Soviets promised to let Poland hold free elections, but soon reneged on that promise. This made the West believe that the Soviets were bent on expansion. This could be seen as the start of the Cold War because the West came to believe the Soviets were a threat to them.
When the war actually did end, there was no longer a basis for cooperation between the various countries. This can be seen as the start of the Cold War because, from that point on, the relationship between the USSR and the West was almost completely contentious. The two sides no longer had a common enemy to keep them together and so they turned on one another.
When I teach about the Cold War, I tell my students that it began when WWII ended. However, I know that this is more convenient than it is accurate. The Cold War had its roots in conflicts from decades before WWII ended. Even so, that is still my answer for how the Cold War began.