How were the Nazis able to make Nazism a mass movement in Germany?

There were three main qualities that led to Nazism becoming a mass movement in Germany: a pervading sense of fear and despair, a charismatic leader, and propaganda.
Following the First World War, the political and social chaos in Germany brought about many violent and extreme political groups. Hitler's National Socialist German Workers' Party (or Nazi Party, for short) was one of them.
Prior to the Great Depression in Germany, this group was perceived as an offshoot of the radical right on the political spectrum. They hardly garnered support, as evidenced in that they only received 2.6 percent of the national vote on May 2, 1928.
During 1930 to 1933, an economic downturn caused people throughout the country to face a worse desperation than ever before. Many citizens were out of jobs and they blamed the parliamentary government coalition for being weak and ineffectual. Widespread fear, anger, and a lack of hope, combined with a sense that the government was unable to manage the crisis, provided fertile ground for the rise of the Nazi Party.
The Nazis promised to turn the country's economic situation around. They assured citizens that Germany would return to its "rightful position" as a world power, that the country would retain its cultural values while resisting the perceived threat of the Communist uprising.
These promises were highly attractive for thousands of people who felt their lives were on the brink of disaster.
Like all mass mobilizations, the Nazis also had a charismatic leader. Adolf Hitler was a persuasive orator who effectively tapped into the anger and helplessness felt by many Germans who were desperate for change.
He organized meetings that looked like military parades; he gave speeches that evoked powerful emotions; and he created subsections of the party that generated a sense of honor, such as the 'stormtroopers.'
Most notably, he created a sense of 'us' versus 'them' among Germans. He directed the population's anger against the Jewish people, the Marxists, and against anyone who the Nazis held responsible for signing the armistice of November 1918 and the Versailles treaty.
Finally, the party relied heavily on propaganda to persuade others of their greatness. They spent huge amounts of money on leaflets, newspapers, and posters with simple messages that encouraged people to join the movement. They held political rallies with large banners and marching bands. They utilized art, books, films, radio, educational materials, and the press to communicate their message. In doing so, they encouraged passivity and acceptance of the fate of millions of people who were part of the 'them' group, all in the name of "restoring order."

Hitler understood that previous far right movements had got nowhere because they were seen as representing the interests of a social elite. He knew that if the Nazis were to succeed, they had to reach out to the masses, to appeal to the broad swathe of middle and working class voters who had always been intensely hostile to the Weimar Republic.
The Nazis did this primarily by playing on the fears and concerns of millions of Germans. The country had been in the grip of a major economic crisis since the onset of the Great Depression. It seemed to many that the old ways of doing things had failed miserably. Germany was an economic basket case, with mass unemployment and hyperinflation plunging millions into poverty and despair. The Nazis were able to exploit widespread disillusion with the existing political and economic system and put forward a radical alternative.
They were always deliberately vague with their plans, making generalized promises which conspicuously lacked detail. But Hitler wasn't interested in the minutiae of policy; he was much more interested in putting forward powerful propaganda messages that could attract the support of the masses. As well as keeping their message simplistic and vague, the Nazis cynically tried to be all things to all men. Whatever your social group, whatever your occupation, wherever you lived, be it town or country, the Nazis had a policy just for you.
Although the Nazi cause did become a mass movement, it's important to acknowledge that they never won a majority of seats or votes in a free and fair election. Turning Nazism into a mass movement may have helped them get a foot in the door of power, but it would take shabby backstairs, intrigue, and a fatal miscalculation by Hitler's political rivals before the Nazis were able to take over Germany completely.