Hitler used violence. Brownshirts went around intimidating and beating up the NSDAP's (Nazi's) opposition and even at times committed murder.

In January 1933 Hitler was able to convince Reichspraesident (President) Hindenburg to make him Reichskanzler (Chancellor). Hitler was able to do this as the Reichstag was fragmented with many parties, none of whom were able to gain dominance. Hitler promised to not support any other party which was a problem as the NSDAP controlled about 1/3 of the parliament. This resulted in multiple elections over short periods of time. Others approached Hindenburg and asked him to make Hitler the Chancellor to end the instability and he finally agreed.

In February there was a fire at the Reichstag. President Hindenburg used emergency powers of the German Constitution to suspend many civil liberties which the NSDAP used to imprison people who spoke out against the NSDAP.

Under intimidation from the NSDAP's Brownshirts in March 1933 the Reichstag voted for the Enabling Act which allowed the NSDAP to rule by decree. It even allowed Hitler to overrule the German Constitution except in limited circumstances. It would expire in 4 years. President Hindenburg signed the act into law. It's important to remember that at any time Hindenburg could have stopped Hitler but he was increasingly becoming a very sick man.

With further intimidation by Hitler's Brownshirts the Reichstag voted to make all political parties but the NSDAP illegal. There was barely any opposition and the law passed.

In July 1934 Hitler set off the Night of the Long Knives. This involved not only the mass arrests and assassinations of opponents to the NSDAP but also the left wing faction of the NSDAP, including SA leader Roehm who was arrested, thrown in a cell and then shot to death. Roehm had been increasingly speaking out against Hitler and it was feared that he would attempt a coup.

In August of 1934 President Hindenburg died. Hitler then combined his position of Reichskanzler and Hindenburg's position of Reichspraesident into one position known as Fuehrer und Reichskanzler. This gave him the power of both Head of Government and Head of State and at this point Hitler become a true dictator. While doing so was technically illegal under the Enabling Act which protected the position of Reichspraesident there wasn't anyone in parliament to stop him.

"Hitler" by Ian Keshaw covers a lot of Hitler's rise to power.

Edit - added a bit about Roehm.

The best book that gives you the everyday experience of the Nazi rise to power is William Sheridan Allen's The Nazi Seizure of Power: The Experience of a Single German Town, 1922-1945. To learn about the kind of person who voted for a Nazi, see Thomas Childers The Nazi Voter: The Social Foundations of Fascism in Germany, 1919-1933

These books surpass anything we can tell you about it. The Nazi rise to power is the sort of topic that cannot be synthesized into a couple of sentences. People's entire careers are focused on not even just that topic, but on ASPECTS of the topic. That's how monumental it is.

I'll try my best to explain.

Hitler originally joined the German Worker's Party in 1919 and later became head of the party in 1923, with it being renamed the National Socialist German Workers' Party (Nazi for short). Hitler attempted to overthrow the German government in 1924. The event known as the Beer Hall Putsch. The attempt failed, he was convicted of treason, and sentenced to prison. He appealed to the judges so they gave him a much lighter sentence. He was released a year later.

Once he was released, he began making reforms to the Nazi party. He was gaining more and more support over the years with the Nazis being the dominant party by the early 1930s. He was appointed as chancellor in 1933 due to the amount of support he got. He quickly gave himself emergency powers, thus making him a dictator. Hitler used his powers to ban all political parties, which made him end up with no opposition.

Now, you're wondering how he appealed to people. Hitler was very charismatic. He appealed to the German citizens at the time because of his promises to fix Germany's problems. You see, after WW1, Germany was hit with various problems. The effects of the Treaty of Versailles, the debt Germany had to pay, etc.

Hitler made very passionate speeches, which people would attend Nazi rallies and meetings just to hear his speeches. He had a way of moving the crowd. Hitler also used a lot of propaganda. Joseph Goebbels, who was in charge of spreading propaganda for the Nazis, did a very good job, After Hitler became chancellor, the Nazis literally controlled everything. Media, entertainment, literature, arts, education, etc. So the Nazis were feeding German citizens with what they wanted them to hear.

tl;dr- Hitler was very charismatic and used propaganda to gather public support.