How did World War II begin?

The crucial word in your question is 'begin' because it does not ask for anything specific, which is why it is such a great question.  It can be argued that World War I set the stage for World War II, simply based upon the Treaty of Versailles.  For all intensive purposes, a treaty is defined as an agreement between hostile nations. However, the Treaty of Versailles, ending World War I was not so much an agreement among nations as it was to be a lesson in punishment for Germany. The Treaty of Versailles held Germany responsible for the war, ordered it to pay reparations, and denied them the right to a military. Combine these mandates with the 1929 stock market crash which was felt just as great in Europe, and the result mired between appeasement and Hilter's march across Europe. The fact was that Germany could never pay the reparations to the nations stated in the Treaty of Versailles. As a result, Germany was left politically and economically fractured. This left the nation vulnerable to those who would use that vulnerability to their own advantage, namely Adolf Hitler. Any nation in turmoil could be blinded by a false the sense of security, disguised in the name of nationalism. WWII began essentially due to the reaction of a nation struggling to make sense of the WWI treaty.  The Treaty of Versailles made the German people feel as if they were 'standing naked on a hill'. Despite WWI and the treaty many Germans felt that Hitler had the capacity, to regain Germany's proper place in world affairs. However, September 1, 1939 Hitler invaded Poland and many of the German people began to question their aliegence to Hitler, keeping those thoughts to themselves for fear of death.

WWII began because it was assumed by the powers that be (after WWI)  that a politically and economically weakened nation would never challenge their power simply because they held (or thought they held) the power.