The peace settlements after World War I (1914–1918) led directly to World War II (1939–1945) because they were too punitive in nature. The victorious nations were headed by three leaders: Woodrow Wilson for the United States, Georges Clemenceau for France, and David Lloyd George for the United Kingdom. Italy, which had fought on the winning side, played a somewhat lesser role.

Wilson had hoped to bring about reconciliation with a moderate peace treaty. But he was opposed by the British, French, and Italian delegations. Those countries had suffered enormous casualties, so a lenient treaty was not politically feasible for their leaders.

The losers of World War I were forced to sign harsh treaties. These nations included Austria, Hungary, Bulgaria, and Turkey. But the most important treaty by far was the one forced on Germany—the Versailles Treaty. It was the most important because Germany was the largest of the defeated nations. The Germans were blamed for starting the war. Also, Germany had to pay steep reparations and cede one-tenth of its territory. Germany's army was limited to 100,000 men; its naval power was also eviscerated.

After Germany's defeat, its government struggled under the weight of reparations. Moreover, its delegates were blamed for signing a treaty which was dictated to them. The unpopularity of the treaty, combined with Germany's economic problems, helped Adolph Hitler rise to power. Hitler's stirring speeches promised to restore German pride. After he became dictator, the outbreak of WWII was practically inevitable.

One of the critical details about the First World War that you should be aware of (especially where this particular question is concerned) is this: when the war ended, German troops were still in France. While the German position had collapsed, they had not yet been pushed back into their own borders.

As if that was not enough, the fall of the monarchy and the armistice came in very close proximity with one another, with the Treaty of Versailles signed by a democratic Germany. (This is an important detail, given that it created a link in the popular mind between the two, and thus poisoned support for the Republic among the German people.)

These factors are important in establishing further context in understanding German attitudes towards the Treaty of Versailles, a treaty which had the intention of punishing Germany and crippling its ability to fight another war in the future. Germany was forced to admit full responsibility for starting the war (The War Guilt Clause) and to make large reparation payments to the victorious allies in compensation. Furthermore, Germany had its military crippled, and it lost a great deal of territory.

The result was to create a great deal of anger and resentment, both about the Treaty and about the end of the war (and Hitler would exploit this anger and resentment in his rise to power).

The terms of the Versailles peace treaty were thought by many—and not just in Germany—to be unduly harsh and punitive. The treaty attributed sole blame to Germany for starting the war, it imposed crippling reparations which could never realistically be repaid, and it severely restricted the size of the German armed forces. All of these measures combined created enormous resentment in Germany. Germans of all political persuasions felt humiliated and unjustly treated by the victorious Allied powers.

Extremist political parties and groups like the Communists and the Nazis were able to take advantage of the increasingly fractious national mood, offering a radical alternative to the nascent system of German liberal democracy established under the Weimar Republic. In particular, the harsh terms of the Versailles treaty breathed new life into a previously discredited German nationalism. The treaty opened up a huge chasm between Germany and the rest of the world, making the country more isolated on the international stage. The pariah status this engendered created the ideal breeding ground for German nationalism. The Nazis were only too willing to exploit the prevailing conditions, presenting the German people with what they saw as a glorious vision of a reborn Germany, throwing off the shackles of Versailles to become a great nation once again.

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). 

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.

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.