How did Hitler conquer so much of Europe in such a short time?
Since you use the word “conquer,” I assume that you are asking about the actual military conquests of various European countries, not the peaceful (though coerced) absorption of places like Austria and Czechoslovakia. The German Army conquered Poland, Belgium, the Netherlands, France, Denmark, and Norway all within the first few months of the war. It then conquered much of the Soviet Union in the early days of Operation Barbarossa in the summer of 1941.
There were a number of reasons why it was able to do so. Let us look at three of them.
First, the French and the British had not been prepared to fight a war. Therefore, they were not really able to do anything to prevent Germany from taking countries like Poland and Norway. They simply did not have the necessary military power ready to go when these invasions occurred.
Second, some of these countries were simply too small to put up any kind of a serious resistance to German invasions. Even in the best of times, a country like Belgium or Denmark would never be able to resist a major power like Germany for very long at all.
Third, the German military had developed its new tactic of “blitzkrieg.” This was a new development in offensive tactics. It combined mechanization of as many units as possible (for fast movement) with close coordination between tanks, foot soldiers, and air support. The armies of France and England were not ready for such tactics, which is why even France fell as fast as it did.
These are the main reasons for Germany's ability to conquer so much so quickly.
If you are writing about the territories such as Austria, Czechoslovakia, and the part of Poland that connected East Prussia, you are asking about the policy of appeasement.
The thought of another war after WWI and the Great Depression was such that, most European leaders would have done most anything in order to avoid conflict. And the reasoning behind the Germans wanting the territory, uniting the German speaking peoples, did not seem unreasonable, at the time. Hitler did this without firing one shot.
Appeasement, the policy of making concessions to the dictatorial powers in order to avoid conflict, governed Anglo-French foreign policy during the 1930s. (History.co.uk)