It depends; who are you talking about? Germany, Hungary, Italy, Yugoslavia, Turks...they all had incredibly different paths toward the relative failures of their democratic systems. I can speak a bit on the German one but it really is just far too general of a discussion to say "THIS is the reason why all these nations democratic systems collapsed"; as if there was some plague out there stopping democracies from happening.

This is a sentiment you'll actually see earnestly applied, especially to the German people, called Sonderweg or the "Special Path" and it's considered total nonsense in 2015 for a reason. However the fact that it was a serious academic movement for quite a while shows the fact that you are not the only one to notice this -- this is not a dumb question and it is one to this day that is not fully answered. Historians "back then" tried to find a unifying reason and this fell into the Sonderweg hypothesis but we now know there is no unifying factor; each one is an individual basket case. Though some themes do repeat: Communist and Socialist uprisings which flourished due to new, untried governments along with economic downturn following the calamity that was the war. Even victors like Italy hardly "won" in a sense.

Looking at a specific case study, Germany, though and we get again a good motif of the reasonings behind the revolutions that spread across the other short lived democracies; notably in the former Austro-Hungarian Empire and Ottoman Turks. The short of it, and this can be a pretty hard pill to swallow for some people, is that the Weimar really had little chance of long term acceptance in the eyes of the people because it was the signatory to Versailles. To say the "Stab in the Back" myth spread like wildfire through Germany would be the understatement of the century; they perceived they were in a winning situation (they weren't) and that victory was in their grasp (it wasn't). Since the Weimar signed the peace treaty they were an illegitimate government in the eyes of many. However that's an explanation for why there was not weeping for the fall of the Weimar, not why it happened. It happened because of right wing collusion and underhanded tactics by the Nazi Party. My favorite passage from Adam Tooze's Wages of Destruction paints this scene perfectly:

"On Monday, 20 February 1933, at 6.00 p.m., a group of about twenty five businessmen were summoned to attend a private meeting in the villa of Hermann Goering, now acting as president of the Reichstag, at which Hitler, the Reich Chancellor, was to 'explain his policies'. The guests were an oddly assorted bunch. The invitees included leaders of German industry, men such as Georg von Schnitzler, second in command at IG Farben, Krupp von Bohlen, who was both head-by-marriage of the Krupp empire and the current chairman of the Reich industrial association, and Dr Albert Voegler the CEO of the Vereinigte Stahlwerke, the world's second largest steel firm.

Hitler launched into a general survey of the political situation...the experience of the last fourteen years had shown that 'private enterprise cannot be maintained in the age of democracy'. Business was founded above all on the principles of personality and individual leadership.Democracy and liberalism led inevitably to Social Democracy and Communism...[the NSDAP] would show no mercy towards enemies on the left. It was time 'to crush the other side completely'... [Hitler] planned to crush the German left and in the process he was more than willing to use physical force. At least according to the surviving record, the conflict between left and right was the central theme of the speeches by both Hitler and Goering on 20 February...Since German business had a major stake in the struggle against the left, it should make an appropriate financial contribution. 'The sacrifice', Goering pointed out, 'would be so much easier...to bear if it realized that the election of 5 March will surely be the last one for the next ten years, probably even for the next hundred years.'

Over the following three weeks [the Nazi Party] received contributions from seventeen different business groups. The largest individual donations came from IG Farben (400,000 Reichsmarks) and the Deutsche Bank (200,000 Reichsmarks). The association of the mining industry also made a generous deposit of 400,000 Reichsmarks. Other large donors included the organizers of the Berlin Automobile Exhibition (100,000 Reichsmarks) and a cluster of electrical engineering corporations including Telefunken, AEG and the Accumulatoren Fabrik...it was the donations in February and March 1933 that really made the difference. They provided a large cash injection at a moment when the party was severely short of funds and faced, as Goering had predicted, the last competitive election in its history."

Fascism can not be separated with the heavy corporatism and industrial collusion that follows and usually precedes it. German democracy failed because German industrialists wanted it to fail. They got political backing from a German Right, represented by the Chancellor Adolf Hitler aka the NSDAP, who viewed the Weimar as illegitimate for the reasons above. Liberals, trade unions, socialists -- all things that were in the interests of "big business" to crush and the Nazi Party, and in general the German Right, had absolutely no issue as well with crushing. The false flag of the Reichstag fire allowed Hitler to brutalize, intimidate, and/or arrest swathes of the Communist and Socialist Party's representatives to disallow them from voting and it shot the NSDAP from having a plurality into a straight up majority thus giving them the power to pass the Enabling Act. The SDP was the only Parliamentary power bloc that stood in the NSDAP's way and they were all jailed or intimidated away from voting against Hitler. That's not a failure of democracy or "the people" rejecting democracy for some other bullshit -- it's flat out underhanded false flag and brutal intimidation toppling an already structurally weak and relatively new power bloc.

TL;DR:

Right Wing parties had heavy resentment...in general. Massive socialist and communist revolutions and sentiments enhanced these ideas. New democratic governments were seen as illegitimate and were barely capable of holding themselves up. A perfect storm to allow constant flip flopping of governmental power (see Hungary, for instance which fell to a Communist Government then back to a Monarchial State). However there is no unifying theory...each case is a specific instance.

Put simply, the western democracies were used to democracy slowly but surely working out solutions to problems. In this case that problem was the great depression. But the new democracies didn't understand this and weren't patient enough. Dictator's can and usually do move faster.