The main way in which the peace settlement after World War I led to World War II was by making Germany feel a strong desire for revenge. The peace settlement also made Italy angry, but this was less important in causing WWII.
After WWI, the Treaty of Versailles was imposed on Germany. The treaty was very harsh. It took away much of Germany’s territory, including all of its colonies. It prohibited Germany from having a military of a normal size and it prohibited the German military from having a variety of offensive weapons. It made Germany admit the war was its fault and it made them pay reparations to the French and the British.
All of these things made Germany very angry. They made the Germans feel that they had been treated unfairly. Because they felt this way, they wanted revenge. When Hitler came to power, he promised to make them strong again. This led people to support him. The desire for revenge and greater power caused Hitler to do things like taking Czechoslovakia and, eventually, invading Poland. This caused WWII to start.
So, the peace settlement after WWI made Germany want revenge and that desire for revenge led to WWII.
During the negotiations for creation of the Treaty of Versailles, Germany wasn't allowed to attend. This ultimately was the wrong choice because the U.S., Great Britain, and France decided to punish Germany rather harshly.
One punishment, which was more emotional than anything, was that Germany had to take 100% of the blame for the start of World War I. Another punishment made Germany pay reparations to the Allies for damages caused by the war. It came out to $33 billion, which Germany only had about a third of that money at the time. Germany also lost about 13% of its land that Germany had previously occuped and all of its overseas colonies. This also allowed them to lose about 10% of their population. Lastly, the German army was restricted to 100,000 men. They also couldn't have an air force, tanks, or submarines. All munitions had to be surrendered, and it demilitarized the Rhineland (the territory between France and Germany).