Well, first off, Europe did not achieve socioeconomic integration after the Second World War. In fact, trade between some nations, such as Czechoslovakia and France, declined considerably in the decade after the war, and Germany was split into two. Yes, we know when reading your question that you really mean to ask why Western Europe integrated economically after the Second World War, which laid the foundation for further European integration at the end of the Cold War. From the perspective of American diplomatic history, the answer lies in this distinction: Europe divided into two camps after the Second World War, each under the tutelage of a superpower. And it was in the interest of the United States to see that Western Europe integrated into a cohesive economic, political, and military ally in the America's global competition with the Soviet Union.

Stepping past the conflicting perceptions of the Truman administration and the State Department on one side and Congress on the other that hindered a quicker American response to the rising Cold War (and also ignoring the role that Truman's perceptions may have played in the rise of the Cold War itself), by March of 1948, a consensus formed in the United States holding that Washington needed to counter Moscow's plans to spread Soviet influence into war-ravaged Western Europe. This led to Congress backing both the Marshall Plan and the North Atlantic Treaty Organization. We typically remember the Marshall Plan for its injection of cash into the European economy, but the plan also required the participating states to take the first steps towards economic integration by removing tariffs and other trade barriers that slowed trade in the region. And despite its role as an enemy of the West just a few years before, West Germany received substantial Marshall Plan funds over the next four years. The threat of Soviet expansion trumped fears of a German resurgence, although West Germany would not join NATO until 1956.

The most important step towards European economic integration that occurred during the Marshall Plan years was not the Marshall Plan itself, however, but a plan proposed by French foreign minister Robert Schuman to prevent future wars between France and Germany by integrating the coal and steel markets that had created a sense of economic competition between France and Germany over the previous century. The European Coal and Steel Community, which came into being in 1951, laid the foundation for the European Union that we know today. The ECSC may have been a European initiative, but it has to be viewed within the context of the wider Cold War. The United States, after all, strongly supported the formation of the ECSC, with the initiative becoming a primary focus of the State Department in 1950-1951. Compare the founding continental European members of Nato in 1949 (Belgium, Denmark, France, Italy, Luxembourg, the Netherlands, Portugal) to the founding members of the ECSC in 1951 (Belgium, France, Italy, the Luxembourg, the Netherlands, and West Germany). Note as well that all of these nations began participating in the Marshall Plan in 1948.

(Largely in response to Western European economic integration, the Soviet Union oversaw similar political, economic, and military integration in the years after the Second World War, but I do not believe this is the area you were asking about.)

In short, the Cold War and the rising influence of America in Western European affairs, as well as a European desire to avoid another devastating war like the two they had experienced over the last thirty five years, spurred Western European integration in the wake of the Second World War.