I'm going to stick to your question of what people thought at the time, and not necessarily the accurate history of the swastika.

Prior to the Nazi party the swastika was not an unknown symbol. It was associated with luck and charms in many parts of the world - however it never had much wide spread adoption as a major religious symbol.

Prior to the rise of the Nazi Party the swastika had been adopted by ex-Baltic soldiers who learned of the symbol from Finland during the Great War. These same soldiers adopted the swastika as their symbol in German Nationalist groups during their opposition to the growth of opposition to the post-war German Republic.

Some anthropologists at the time also believed the swastika to be of Aryan origin, and that it's spread followed the spread of Aryan people across Europe and Asia. They dated archaeological finds to make the 'Danubian Culture' older than the other south-eastern cultures who also used swastika and to establish a kind of Indogermanic origin for European and Asian culture to spread from. Again, it is being associated with German nationalism and primacy.

Adolf Hitler recognized the powers of symbols and pushed the swastika strongly, while at the same time carefully preserving and cultivating it's use. He was very aware of the symbols use among German nationalists already and association with Aryan culture. Under Hitler you could not use the swastika commercially or profit from it's depiction. In time the swastika came to symbolize Germany and German people. It was medieval in a way, and Hitler had his own personal standard and heraldry based on the swastika.

The flag and symbols of Germany were not replaced wholesale either. The flag of Germany was both the historic Imperial tri-colored flag (different than the tri-colored flag of the German Republic) and the swastika for a three year period until the swastika replaced it totally in 1935. It was retired because Goering claimed it represented a bygone era and that it could be used by reactionaries to oppose the new Germany. To move forward a new symbol was needed for Germany, and conveniently they had one ready. By this time the Nazi party and Germany are coming to be seen as the same thing in many people's minds. This official merging would only solidify it.

Similar to the flag the traditional icon of the German Eagle was used until 1935 when it was replaced by a stylized Eagle atop a swastika. In this case Hitler is co-opting the black eagle motif and modernizing it and merging it with the swastika. This symbol is evolving under the Nazi party, not being replaced. Other traditional symbols were co-opted. The Iron Cross was still used and was of Teutonic origin through along with everything else it was modernized.

It's also important to realize that Germany during the 20s and 30s was undergoing tremendous change already. The flag and symbols of Germany were constantly changing every few decades anyway (though not as drastically as to the swastika) so it was not as shocking as if the USA changed it's flag. People were also upset and dissatisfied with the German Republic, so discarding it's symbols was not seen as a bad thing by all.

Germany was also not a free society. Dissenting opinions were oppressed and the press was strictly controlled. To oppose or publicly resent the swastika could have been, at a minimum, politically dangerous. You never knew who was reporting to whom. It's also important to not equate the Nazi party with political parties like we have in modern times. The Nazi party came with its own secret police and military branch. It was much more than just political.

So the swastika did not suddenly replace everything. It had been growing in visibility since 1918 and became associated with German nationalism. With Hitlers growing power, and a growing bad taste for prior German governments it was not seen as such a bad thing to replace symbols. By 1936 the Nazi party, Hitler, and German Nationalism was tremendously popular and the swastika was ubiquitous. To the untrained eye Germany seemed to be doing well.

Edit: I'm not aware of the swastika appearing on much military hardware, so I doubt there was much resentment there. The Wehrmacht used a stylized Iron Cross as their symbol almost exclusively.

Sources:

The Swastika - The twilight of a universal symbol by W. G. V. Balchi (Interestingly this was written during the war, and the author laments the impending suppression of the symbol)

The Swastika; Its History and Meaning by John Prince Loewenstein

There was a lot of controversy regarding the German national flag during the Weimar Republic. The new Black-Red-Gold of the Republic, which went back to the revolution of 1848/49 and the first German national assembly, with connections to the students movement and the national wars of liberation against Napoleon, was hated by nationalists and right wing forces, who favoured the Empires black-white-red, while communists would have rather seen the red flag wave over Germany.

Often disparaged by right wing, nationalist, monarchist and revisionist groups, for example as 'black-red-mustard', it was for many a hated symbol of a hated system that would hopefully be gone sooner or later. These groups largely favoured the old black-white-red of the old Empire, as on this DNVP (German National People's Party) poster from 1924. This DNVP poster from 1925 shows how the republican flag was viewed by the right, the proud Prussian eagle contrasted with the scrawny, sad Reich-Eagle.

On the other hand, the black-red-gold flag of the republic was offensively used by pro-republic and centrist groups to distance themselves from extremists of the left and right fringe. This election poster used by the DDP (German Democratic Party) for the 1928 election exemplifies this very well: The Republican flag, here named 'Reichsflagge', Empire Flag, is shown specifically in contrast to the flags other parties, from the right as well as the left (note the DNVP flag with the Swastika!) would presumably favour. Here is another nice example, a DDP poster from 1928, which shows the black, red and gold as a veritable shield against communists, national socialists and monarchists alike ("Clean up the Reich! Vote German Democrats!"). On this SPD poster from 1928 we have similar symbolism, with the black-red-and gold of the republic combined with the red flag ('liberty') flying, and protecting the tower. This example from a centrist coalition for the candidate Wilhelm Marx (Reich chancellor 1923/24 and 1926-28) makes a direct comparison to the revolutions of 1848 and 1918 ("What the fathers thought of in '48, the grandchildren completed in '18! The Banner, which Grimm and Uhland unfurled - you want to betray it? : Black-Red and Gold!! Vote for the candidate of the People's Block: Wilhelm Marx.")

The Swastika itself was already in the 20s often seen as a symbol used by nationalistic groups, appearing for example on this election poster of the DNVP from 1924 and this NSDAP poster from the same year. Since the beginning of the 20th century, it had already been used by the völkisch (nationalistic) and antisemitic movement and later the Freikorps after 1918 - Here are members of a Freikorps during the Kapp-Lüttwitz-coup in 1920 with Swastikas painted onto their helmets. Therefore, it was a very familiar symbol for the German right wing, and easily accepted.

In turn, the Swastika was identified as symbol for the political enemy, for example in this SPD-poster ("The worker in the realm of the Swastika -therefore, vote list 1, Social-Democrats!") - note how he is tied to the bars with black-white-red cord. Here (again SPD, 1932) it is set among other symbols of anti-republican forces, communists and monarchists. The three arrows are a symbol of the Reichsbanner Schwarz-Rot-Gold (Reich Banner Black-Red-Gold), a para-military organization dominated by the SPD which had as its goal the protection of the Republic against extremist forces.

Since Black-White-Red was so popular, some concessions were made, like on the [merchant flag of the Weimar Republic](http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Flag_of_Weimar_Republic_(merchant).svg/800px-Flag_of_Weimar_Republic_(merchant).svg.png), where the republican flag only plays the smallest role in the canton; sames as in the [Reich War Flag](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Flag_of_Weimar_Republic_(war).svg/220px-Flag_of_Weimar_Republic_(war).svg.png) from 1920 on. Sometimes, nationalistic groups hoisted the Black-White and Red flag in defiance, for example after Hindenburg was elected in 1925. The German Football Association (DFB), who played (and still play today) in the old Prussian colours of Black-White, often hoisted the old flag before games instead of the Black-Red and Gold. This so-called Flaggenstreit (flag strife) escalated in 1926, when Reich Chancellor Luther tried to introduce a motion, which would have allowed German embassies and government representatives to hoist the Black-Red and White merchant flag with Black-Red-Gold in the canton when overseas. This led to furious protest by Republicans, and Luther finally had to resign after losing a vote of confidence introduced by the DDP.

So when the National Socialists took over in 1933, there was a situation in which there are no commonly accepted traditional national symbols, not even the eagle, since there were several different forms, some used by the republic (largely the same which are in use today). Immediately in 1933, the old Black-White and Red flag was adopted as the official National Flag. Many times it was hoisted alongside the Swastika flag - which at that time was still only the flag of the NSDAP, not Germany, but some resisted this, insisting on only using the black-white and red, such as the mayor of Leipzig (and later resistance member) Carl Friedrich Goerdeler; which led to a decree by Hindenburg that both flags were to be hoisted alongside. In most cases, though, this duality was not only tolerated, but actively embraced. Only in 1935, after Hindenburgs death, did the Swastika flag become the sole National flag and ensign of the German Reich (a catalyst for this might have been an incident in New York, where the Swastika flag flown from the SS Bremen was torn off and vandalized - the complaint by the German ambassador was answered by a reference to the fact that it was only a party, not a national flag).

However, the black, white and red of the Swastika flag were not intended to be representations of the old monarchistic colours, even though Hitler looked favourably on the connections these colours had to his time in the military (cf. Mein Kampf, p. 551 ff.); but the assocation with the monarchistic colours placated nationalistic and revisionist groups who were not part of the National Socialist movement. The revolutionist break with the old Germany was very important in the symbology of the Swastika flag and other national socialist symbols, which is why red is so dominant, and why other symbols such as the eagle appear in highly stylized and changed form.

The military didn't use the Swastika to paint onto their vehicles (they did use the Swastika in the War Flag and some other insignia), but rather the Balkenkreuz, a simplified form of the Iron Cross (in the beginning sometimes even just as a white cross). However, a red cloth with a centered white disc and black swastika was used by the Wehrmacht as a "Fliegertuch", 'aviator cloth', which, draped across the hood of vehicles or laid out on the ground, aided in friend-or-foe identification at the front from above which otherwise would have been very difficult considering the swift-moving and often confused front line in the war. This, however, was always referred to as Fliegertuch, not as national or party flag. In fact, the military apparently was quite happy to have an identifying symbol with such a stark contrast and high visibility.

There was resistance to the new symbols, such as Goerdeler did, but acted upon only sparsely and isolated. The important thing was that the Swastika and other NS-symbols stood for a revolutionary break and a New Germany, they didn't need tradition to recur on. For many, it was still the symbol of the enemy, but effective resistance was hard and increasingly harder when old socialist and communist organizations (such as the Reichsbanner) and underground networks were eliminated. Opposition to NS power was weak and crumbling from the beginning, and with it any possible resistance towards the new symbols of the Reich. Kurt Tucholsky, a German essayist and satirist, put it thus: "You do not whistle against the ocean."

Then of course you have the view held by some NS-ideologues such as Rosenberg that the Swastika was, in fact, a traditional aryan-german symbol.

In "The Third Reich, A New History" by Michael Burleigh, there is a potential insight into why it was accepted as well as it was:

The Nazis recolonised the red flag, setting within it the swastika symbol, which in turn they hijacked from earlier Völkish sects and the Free Corps, cunningly incorporating the imperial black and white into the colour scheme.

The paragraph previous to that explained how the Nazis were appropriating the imagery of competing movements, such as the Marxists:

They also shamelessly plundered the literary and visual repertory of their "Marxist" opponents in order to sow confusion and exploit similar aspirations and resentments.

They tried to be all things to all people and the symbolism was part of it. Let's not forget also that as time went on it became more and more dangerous to oppose the regime, so the combination of these factors led to the acceptance, willing or otherwise, of the swastika.

Edited for spelling