What caused World War II to break out in 1939?
As outlined in pohnpei's answer, Germany's expansionist aims in Europe were the immediate trigger of the Second World War, when Britain and France stepped in following the German invasion of Poland.
Of course, it should be remembered that German expansionism had its roots in the outcome of the First World War, when Germany was stripped of land and power by the victorious nations, chiefly Britain and France. There was simmering resentment among many in Germany about how the country had been treated, and Hitler was able to exploit these grievances as he talked up the cause of German nationalism. The ignominious fate of Austria - another Germanic country and formerly the seat of a centuries-old European empire - in the First World War also played into this.
The antagonism between these Germanic powers and Britain and France really went back to the nineteenth century when the ascendant German kingdom of Prussia unexpectedly defeated France in war. France and Britain were long-standing imperial powers in Europe (and had often bitterly fought each other) but in the later nineteenth century Germany began to seriously challenge them both. This rivalry between major European nations and the concomitant arms race was in large measure responsible for World War I, and the after-effects of that war carried over into World War II.
There were several factors that caused World War II to begin in 1939. Throughout the 1930s, Germany, Japan, and Italy became very aggressive in their desire to control land. Japan invaded Manchuria in 1931 and China in 1937. Little was done about these invasions. Germany began to build up its military in the 1930s, which was in violation of the Versailles Treaty. Germany also violated the Versailles Treaty by moving its military into the Rhineland. Italy invaded Ethiopia in 1935. In each case, little or no action was taken against these countries.
These aggressive actions continued in the late 1930s. In 1938, Germany annexed Austria. Germany announced it wanted the Sudetenland, an area in Czechoslovakia where many Germans lived. The Allies made an agreement with Germany that gave Germany this land in return for a promise to take no more land. In 1939, Germany broke this agreement, called the Munich Pact, and took the rest of Czechoslovakia. The Allies told Germany if Germany invaded any other countries, it would lead to war. In September 1939, Germany invaded Poland. Great Britain and France then declared war on Germany, leading to the start of World War II.
The immediate cause of the outbreak of WWII in Europe in 1939 was the German invasion of Poland. England and France had guaranteed that they would protect Poland’s borders as they were in 1939. When Germany invaded, France and England kept their promise and declared war.
The broader cause of the war was German aggression and French and British appeasement. Germany, under Hitler, had been trying to expand and create what it saw as a Greater Germany that included all ethnic Germans. The French and British had simply allowed this expansion, not doing anything when Germany annexed Austria and then took Czechoslovakia. Since the allies did not act, Hitler was emboldened to take Poland as well, not believing England and France would go to war.
However, England and France decided they could no longer back down and they went to war when Germany invaded Poland.