What facilitated the spread of Communism after World War II?

There are a number of factors. First of all, the Soviet army had boots on the ground in several East European countries. This gave them enormous leverage in any future political settlement, as was shown quite decisively at Yalta. Long before the end of the war, Stalin had already prepared quite carefully what he was going to do in Eastern Europe. As Soviet troops occupied each country, an apparatus of repression went with them. Secret police, propagandists, and political commissars all followed the train of Russian soldiers to get ready to establish Soviet domination as and when required.
Communism tends to thrive in conditions of extreme economic distress, especially in capitalist countries. Communist parties offered the prospect of a better economic system, a radical alternative to the prewar system of free market capitalism and Fascist corporatism. Europe at the end of World War II was in a state of economic exhaustion; in this environment it is not surprising that Communist parties found fertile soil in which to grow. In Italy, for example, the PCI under Togliatti came a close third in the 1946 General Election, with nearly 20% of the vote. Indeed, it was the success of Communist parties in Europe that provided the main impetus for the Marshall Plan, an attempt by the United States to revive the continent's shattered economies.
The case of Italy also provides another example of why Communism spread so rapidly after the war. Communists in a number of countries had been at the forefront of anti-fascist insurgency. This had greatly enhanced their reputation in the eyes of many. The link between Communists and national liberation struggles can be seen most clearly in China, where the Red Army under Mao had fought not just against Nationalist forces but against the Japanese invaders.
https://carleton.ca/ces/eulearning/history/europe-after-wwii/the-reconstruction-of-europe-in-the-shadow-of-communism-external-pressure/