How did the peace settlement after World War I lead to World War II?

The peace settlement from World War I led to World War II because it left Germany angry and eager to abrogate that settlement. The anger Germans felt over the settlement led them to support radicals like the Nazis, who then led them into World War II.
The peace settlement after World War I, particularly the Treaty of Versailles, angered Germans tremendously. The treaty required Germany to give up all of their overseas empire and some of their territory within Europe. It even banned them from stationing troops on some of their own soil. It did not allow Germany to have a true military, banning anything the Allies deemed to be offensive weapons. The treaty forced Germans to give France the products of some of their mines. It required them to pay reparations for damages caused by the war. Finally, it required Germany to admit the war was their fault. All of these provisions angered and humiliated the German people.
Because the German people were so deeply wounded by the peace settlement, they resented it and wanted to change or destroy it. This made them more likely to support politicians who were radical if they promised to undo the unfair settlement and make Germany great and respected again. Hitler and the Nazis promised to do just this. They promised to disregard what the Allies wanted and restore Germany to its rightful place as an important and respected European power. Because of this, Hitler received enough support to become Chancellor of Germany.
Once Hitler was in power, he was able to do whatever he wanted. This included expanding German power and territory until he invaded Poland, which prompted the Allies to get involved, beginning World War II. The peace settlement led to German anger, which let Hitler and the Nazis rise to power, which caused World War II to begin.