Well Fascist Italy's record of conquest is abysmal to be blunt. So I'll concentrate on Germany.
Germany's territorial demands from 1933 -1939 were to a large degree legitimate. HOWEVER, the passivity of the Allies to German rearmament, and the de facto recognition of it with the Anglo-German naval agreement of 1935, encouraged German aggression in Central and Eastern Europe. There was no doubt that Hitler wanted to destroy the Soviet Union, and thereby create "lebensraum" for the new German Empire (or "Reich"). He was willing to cut a deal with Poland -who had defeated the USSR in 1920, to allow him to station troops and use Poland as a springboard into the USSR. Poland's gov't distrusted the Nazis almost as much as the Communists refused. Speaking of the communists, one of the reasons the western allies were reluctant to contain Hitler was some of the elites some him as a buffer against Soviet expansionism. Militarily Poland had one of the largest standing armies in Europe (larger than the German army until 1936), so they didn't feel threatened.
Militarily, the reason for German military superiority from 1939 until 1943 (or '44) was threefold: doctrine, leadership, and equ;ipment.
Doctrinally, the Germans had appropriated some ideas originating in France and the UK about operating tanks in mass, with infantry, artillery, and engineers. You might have heard the term "blitzkrieg"-this is what it means. Probing for weak points, then exploiting them with massed attacks. (BTW "blitzkrieg" is a German propaganda term, even today German officers use "Combined Arms" to describe operations in WW2). They took it much farther, and added in the idea of what is now called "close air support". This doctrine was revoutionary in its day, and overwhelmed the WW1 indoctrinated armies they opposed.
AFA leadership, the Wehrmacht instilled in its junior officers and senior non-commissioned officers the idea of taking initiative and acting in the absence of orders. The training programs for both were much tougher than the Allies-even up to 1944. The troops had confidence in the leaders, and in their equipment, though they were often outnumbered (as in France, North Africa, and frequently in Russia).
German equipment in many ways was technologically superior to the Allies, better machine guns for greater firepower at the squad level, armored carriers for infantry and engineers to accompany the tanks, more flexible artillery. The single most important equipment factor was: two-way radio communication in their tanks. NO other army had this until 1942 or even later in the case of the Soviets. Combining this instantaneous communication with the initiative of junior leaders, well you read the results. Keep in mind that in France, the Germans faced more and technically superior French armor.
Me. Military Historian.