I agree with both answers, but would add that AQWF also was an embarrassment for Germany. The soldiers were abandoned for all intents and purposes, left to use machinery that was actually killing their own soldiers: shells were being fired from guns that were so ineffectual and/or broken down they would land on the German soldiers and not even make it to the enemy territory. The novel describes the deplorable conditions that the soldiers endured and ridiculed the commanders. And, don't forget the sub-human conditions of the hospitals: doctors hacking limbs because it was quick and easy; soldiers suffering needlessly due to a lack of proper pain medicine; unsanitary conditions in hospitals; inhumane attitudes and treatment of soldiers by doctors and nurses.
The Nazis rejected All Quiet on the Western Front because the book clearly showed the brutality of war and its effect on both soldiers and the civilians left behind. The young Germans enlisted with eagerness and enthusiasm with a desire to proclaim their support for the Fatherland and of a master race. The realities of war were in sharp contrast. Remarque’s most clear example of that came in the scene where Paul was forced to stay in a foxhole with a French soldier whom had just been killed. He realized that he bore the man no enmity. He believed that the leaders should be the ones to go in a boxing ring and fight it out and not the ordinary soldiers.
The Nazi’s needed more and more soldiers and were even taking very young teens as their situation became more desperate. All Quiet on the Western Front presented the war in all of its graphic horror and would disillusion the young men or parents of young men.
Because the book tells of a generation of men who were destroyed by the war (WWI). Protagonist-narrator Paul Baumer traces war's effects on him and his classmates who patriotically enlisted en masse to fight for Germany, their Fatherland. Once into battle, however, they lose their patriotic illusions as well as their limbs, eyes, and even lives.
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I agree with all these answers, but would like to add that Hitler was also a soldier in WWI and that part of the appeal of Nazism was to regain Germany's supposed "lost glory" and strike back at those Hitler believed had "stabbed Germany in the back" and prevented her from winning the war.
In short, Germany was humiliated by the war and many people were left in financial ruins in the wake of the power vacuum and the heavy reparations thrust upon Germany. Hitler seized upon such notions and claimed that Germany shouldn't have lost the war. Certain people (read: Jews, Communists and other so-called undesirables) had lost the war for Germany by giving up and surrendering.
Remarque's book was thus the antithesis of what Hitler was trying to accomplish. The Nazis certainly didn't want anyone reading a book that spoke out against war and considered the human cost of such an undertaking.