The short of it was that he wasn't. He was appointed in order to keep his rising ambitions in check. The Nazis had lost seats in the last elections, but they were militant and were more willing to edge towards open revolution compared to the communists whom were gaining seats in the German parliament.
President Hindenburg was convinced that Hitler as Chancellor would placate his ego, temper the hotter heads in the Nazi party, slow the assent of the communists, and hopefully he could be controlled and "guided" by the Wiemar government for their benefit (perhaps returning back to the old imperial order of the kaiser) or at least to buy some time as it was on the verge of collapse.
Of course it didnt work for them, and Hitler basically put his foot on the gas towards running Wiemar Germany over the cliff.
Source: Inside the Third Reich: Memoirs by Albert Speer
The rise of the Nazis within the German parliamentary system (Reichstag) is a complicated story that plays out over several years and elections, but you could summarize it something like this:
Things were extremely bad in Germany in the 1920s-1930s, due in part to the terms of the Versailles treaty that had ended World War I. The result of this badness was incredible political and economic instability. There were small revolutions being fought, multiple attempts to overthrow the government, multiple people showing up and declaring they could fix all of Germany's problems if you just gave them a hand.
So in the 1930s you have a vying between two very violent groups — the Communist Party of Germany (KPD) and the Nazi Party of Germany (NSDAP). Both were advocating revolutionary changes. You also had the Social Democratic Party (SPD), which was left-leaning but not as far as the Communists, and you had the more traditional and conservative parties (e.g. Centre Party).
The Nazis started gaining serious seats in the Reichstag in the 1930 election, going from about 3% of the entire Reichstag to 18%. In July 1932 (a special election held by existing Chancellor von Papen to try and unsettled the center-left coalition in the Reichstag) they got about 37% of the seats (not what von Papen had wanted), making them the single largest party (but not a majority). They lost seats in the November 1932 election (putting them at 33%), but were still the largest single party represented.
Why did people support the Nazis in these elections? In part because they seemed to be offering an alternative to the Communists. If the choice was between Nazis or Communists, as many people in Germany saw it at the time, then that's a tough choice. Both were violent and intolerant of dissenting opinions. The Nazis promised to restore Germany's original strength and emphasized their German heritage. The Communists promised to overturn everything and looked proudly to the Soviet East. The other more middle-of-the-road parties seemed to not be able to accomplish anything to improve the country. As happens time and time again, when people sense that the status quo isn't working they start looking for alternatives, and when the loudest alternatives are extreme ones, many start looking to the extremes. To many, the Nazis looked like the better option — some radical changes promised in upsetting the existing political order, but ultimately not many changes with regards to the existing economic order (unlike the Communists).
It is worth remembering that up until this time, the Nazi stance towards Jews had been mostly rhetoric, and that World War II was many years away. It was easy for many Germans to think that this rhetoric might just stay rhetoric. That the Nazis might ultimately committed atrocities is easy to see in retrospect. But people tend to suspect that political rhetoric is necessarily overheated. The German people, in any case, had long-standing suspicions and animosities towards the Jewish people, and many were willing to believe that maybe they were in some way responsible for many of their ills. Goebbel's once remarked that you did not need to convince the bulk of the German people hate Jews — you just needed to keep them from caring enough to think about them too deeply, thinking that there must be some reason they are being persecuted. (I am paraphrasing.)
Franz von Papen, the existing Chancellor and an adviser to President Hindenberg, afraid of Communists, somewhat sympathetic to the more conservative leanings of the Nazis, thought that if they made Hitler Chancellor (and Paper would be Vice-Chancellor), they could then manipulate them to support their own interests (namely, the more conservative ones, against the Social Democrats and the Communists). Hindenberg was convinced, and they made Hitler Chancellor. Chancellor was not a directly elected position — it is somewhat like the position of Prime Minister elsewhere, but appointed by the President — and under the Weimar years was actually a weak political office. So the reasoning here is, take Hitler, give him this honorific position, and make him work "within" the system. In theory, he'll become part of the system, will start to compromise, and will take all of those Reichstag seats and use them to pass legislation that the conservative elements of Germany thought were a good idea, against the revolutionary sentiments of the Communists.
But it didn't work out that way, obviously. After a mere month Hitler had utterly transformed the position, used it to suppress the Communist votes violently, and passed emergency "enabling" legislation that allowed him to bypass the Reichstag altogether. At this point he had become a dictator in all but name; after Hindenberg died in 1934, he declared himself Führer (Leader), making himself one in name as well.
As you can see from the above (which smooths over a lot of detail, despite its length!), there were democratic aspects of the Nazis coming to power, to be sure. But it should be remembered as well that much of the support for the Nazis became about who they were opposed to (the Communist Party) — and when it comes to choosing between Hitler and Stalin, there are no right answers. Ultimately the rise of Hitler to Chancellor and from there to Führer was about non-democratic, internal politicking by von Papen and Hindenberg, who made dramatically terrible mistakes in judgment about Hitler's capabilities and ambitions.
There are many books on this, but I am partial to Turner's Hitler's Thirty Days to Power: January 1933, which covers this as well as the way in which Hitler took a weak and calculated political appointment and used it to utterly dismantle Weimar democracy. Also, it kind of sounds like a self-help book ("You too can become Führer, in only a month!") which I find darkly amusing.