The simple fact of the matter is that Germany was never in as bad a position after WWI as is commonly believed.

The Hyperinflation, which stemmed partially from wartime financial decisions of the German government and partly from deliberate attempts to sabotage reparations, was largely under control by 1924. Thanks to the Young-Dawes Plan, Germany now had access to foreign loans and enjoyed immense economic prosperity from 1924-29. Following the Great Depression, the German economy was stabilized by the efforts of Hjalmar Schacht, utilizing Keynesian methods. Hitler used the opportunity to begin a massive armaments programme, part of Goering's Four Year Plan, which placed Germany in an advantageous position in 1939 compared with it's neighbours.

By 1933, the population of Germany was c. 65 million, the same as the much larger German Empire in 1914. The incorporations of the Saarland, Austria, Sudetenland and Czechlands into Germany between 1935 and 1939 ensured that the population in 1939 was much larger than in 1914, and Germany had a much larger industrial base, which was geared towards war.

It also helped that the German military had hardly been curtailed by Versailles. 115 000 men was still a large army and to this could be added the 'Black Reichswehr', a clandestine organization that kept tabs on the right wing Freikorps paramilitaries, so as to call them up in case of war. The General Staff was dispersed through the so-called Troop Office and various other organizations, such as the Reichs Archiv, where former officers wrote apologist histories of the First World War. After the Treaty of Locarno in 1926, Germany began cooperating with the USSR: German officers trained at Soviet Academies, German pilots trained in the USSR, and German aircraft and tank designs were tested in the USSR. Hans von Seeckt, essentially the chief of staff of the German Army, sought to build up the Reichswehr as a professional force which would form the nucleus of a new German Army in the future. All of these efforts ensured that when conscription was reintroduced and the formation of the Wehrmacht was announced, Hitler's generals did not need to start from scratch.

Poland was no match for Germany militarily in 1939; they found themselves in a two front war with the Germans and the USSR, with attacks all along their border from the Baltic to Slovakia. No support was forth coming from Britain or France, but this did not prevent the Poles from putting up staunch, heroic resistance. The Polish campaign cost the Germans c. 50 000 casualties, 1/4 of all their tanks, 285 aircraft lost and c. 280 more damaged.

France was attacked half a year after the Polish Campaign. In this case, the Luftwaffe was able to attain air superiority, and so the movement of the German Army Group A through the Ardennes went largely undetected. The French and British were expecting the main thrust through Northern Belgium as in 1914, and were surprised when they had a large German mechanized force in their rear, and had no strategic reserves. Even still, the Battle of France was a costly affair for the Germans: c. 158 000 casualties, 795 AFVs destroyed, c. 1300 Aircraft lost.

The Battle of Britain, suffice to say, was a failure. The German Navy was reeling from Norway, and the Luftwaffe was inefficient at Anti-shipping operations, so the Royal Navy was a formidable barrier to invasion. The German army lacked the specialized equipment and the training to carry out an amphibious landing. Above all, the RAF was NEVER as close to defeat as the Germans OR the British believed. The losses the RAF suffered could be replaced relatively easily, but German aircraft production lagged behind, and the losses the Luftwaffe suffered were unsustainable. What comes to mind is the dark humour of Luftwaffe pilots, who announced the approach of British aircraft by saying, "here come the last of the Spitfires, AGAIN!" Above all, the losses incurred by the German bomber force in the BoB and the Blitz left it hamstrung for Operation Barbarossa.

To make a long story short, Barbarossa was a failure; spectacular gains were made and losses inflicted, but the USSR survived. From 1942-43 the Red Army gradually got back on it's feet, and by 1944 the war in the east, and in general, was lost for the Germans.

After WWI, Germany was economically devastated. There were restrictions placed on their military strength and reparations that were forced to be paid with the stipulation that if they were not paid, Britain and France would seize German lands and divide them as a result of secret treaties not made public after WWI. (On a side note, the US would pay the reparations for Germany in an effort to maintain peace when Germany was unable to pay due to rapid inflation). Germany was in chaos with returning soldiers organizing into different revolutionary sects and open conflict on the streets. The Nazi party organized and specifically targeted the poor and disaffected - poor people who could not afford to eat and middle class families that had lost their savings. Only the mega-rich were able to make any money or could afford to live relatively normal lifestyles. Hitler and his Brownshirts (who were like the security force during rallies and such) presented to the German people not only organization and a greener pasture ahead, but promised jobs and economic stability, earning popular support enough that he was given a position in government in an effort to control him. Despite the limitations on the military, after gaining dictatorial power, focused most of German production on military strength because he knew what he wanted. He had a plan to reconquer lands that he felt had no right to exist such as Czechoslovakia and Poland. He felt that these countries were part of Germany and should be returned to the Reich. They produced tanks at an unprecedented level. Unexperienced Germans "volunteered" in the Spanish civil war to learn flying techniques and maneuvers and utilized this experience in the air. France was organized defensively and when the Germanys invaded, they simply bypassed French fortifications and cut off those who were fortified. The speed and efficiency that the Germans exhibited with their mechanized divisions was unorthodox and devastating. When you're asking how it happened, you have to take into account the goals of Hitler, the reluctance of Britain and France (who were arguing amongst themselves at the time) to get into another war, the deception and strategy of the Nazis in professing peace while at the same time manipulating countries like Czech and Poland into submission, their u-boats which were very successful early on, the blitzkrieg strategy, etc. Also, keep in mind that Russia was a German ally when it all started. Russia helped in the invasion of Poland and was sending supplies to Germany up until the day before Germany invaded Russia. Many believe that had the Germans continued onto Moscow instead of diverting to focus on supplies and pausing long enough to allow logistics to catch up, the Germans would have handily defeated the Russians by toppling their government and maybe have been able to resist American and British forces. There are so many what-might-have-happened-ifs, but I don't think you're asking about those. To answer your question of how it was possible: The Nazi party devoted almost ALL of their focus on developing their military with the purpose of invading their surrounding countries and their equipment, strategy, and tactics were far superior when the war began.