Note that this is partially based on an earlier answer by me, /u/vonadler, that can be found here.
First some of the basic facts. One needs to remember that while the Swedo-Norwegian Union had a common security, trade and foreign policy, everything else was divided into two parts. Norway had its own parliament, its own budget, its own tax system and state income, its own army, its own navy, its own merchant navy, its own courts and justice system, its own police, its own constitution and laws and its own system of education. In many ways, Norway was already an independent country.
From 1884, King Oscar II was forced to accept parliamentarism in Norway, as the Norwegian radical liberal party, liberal party and agrarian union had joined together in grand coalition called Venstre (left) party and initiated what was called the 'fist policy', ie a confrontational policy against the union where Norway demanded more autonomy or even independence. This slightly Johnny-come-lately during the era of nationalism was nevertheless popular and started to gather broad support in the Norwegian populace - including people earlier focused on more internal issues, such as language, culture and education. The 'fist' policy started to centralise around Norwegian consulates. The large Norwegian fishing, whaling and merchant fleet often needed help in foreign ports with issues around permits, tolls, export dues or in cases where sailors were arrested for being drunk and disorderly and the Norwegian parliament started to demand independent Norwegian consulates so that the government could support one of the most vital trades in the country. However, the Swedes and their supporters of the union in the Høyre (conservative) party resisted. During the early 1900s, it looked like a compromise would be possible, and the conservatives won the election of 1903, promising to resolve the consulate question by negotiations. The Norwegian parliament voted through a new law as basis for the negotiations in Spring 1904. In November 1904, the Swedish response came, and proved completely unsatisfactory, both to the nationalists and the unionists in the Norwegian parliament. Sweden demanded to keep controlling the foreign policy completely. Now the conservatives started switching sides from the unionist faction to the nationalist faction and started demanding independence too.
The King, Oscar II, refused to sign the law the Norwegian parliament had voted through on the 25th of May 1905, and the Norwegian government resigned in response. As neither the conservatives nor the lefts were willing to form a government, King Oscar II and his representatives were unable to find willing Norwegians to staff a government. In response the Norwegian parliament declared the Union dissolved on the 7th of June 1905, citing the inability of the King to appoint a government according to the constitution. Norway held a plebiscite on the 13th of August, where an over-whelming majority of the eligible voters (men over 25) voted for independence. While there certainly were cases of influencing voters - the nationalist factions arranged for campaigns where bands would play the national anthem outside the polling places, the Norwegian flag was frequently flown both outside and inside the polling places - in many cases with Ja (Yes, to independence) embroidered in white on the red and blue of the flag and strong rhetoric called anyone who voted against independence a traitor, the result was still pretty much legitimate. 85% of the eligible voters turned out, and 99,95% voted for independence. While women did not have the vote at the time in Norway, in addition to the 368 208 votes for independence, womens' organisations collected over 250 000 signatures from women in favour of independence.
The campaign before the plebiscite saw a massive nationalist surge, where pretty much everyone and every organisation turned out to be in favour of independence. All the political parties in the parliament were for, the church was for, all the newspapers were for, as well as any other national organisation.
After this negotiations started in Karlstad on the 30th of August. Negotiations continued, and a military solution of the conflict seemed likely in September as the Swedish conservative government was not willing to accept Norwegian independence.
The international community did not want to see war over the issue. The British, while unwilling to actually commit anything officially, were in favour of Norwegian independence, while the Germans, who had earlier indicated support for Sweden, issued only sympathetic statements for the position of King Oscar II, while Kaiser Wilhelm II in private called the Swedes "Schlappschwänze!" (which translates roughly as cowards) for not having intervened militarily already.
On the 18th of September, Norway issued a partial mobilisation and moved 22 500 men to the border. The Swedes replied with their own mobilisation of roughly 100 000 men and pulling together the Swedish navy on the western coast of Sweden - including Sweden's first submarine.
However, the Swedish army, made up of a majority of conscripts since 1901 were less than enthusiastic about going to war against a brethren people and while it was much larger than its Norwegian counterpart, the level of technology were about even - both sides made use of modern bolt-action rifles, modern light artillery, obselete medium artillery and a smattering of machine guns. However, Norway is mountainous and easy to defend, especially by a motivated force defending their homeland. The Norwegian army had since the 1880s build fortifications along the border and trained for a potential war against Sweden, efforts that had increased since the 1895 crisis between the two countries.
Another important factor was the Swedish public opinion. Most Swedes were not willing to shed blood to keep the union going, especially against a brethren people such as the Norwegians. The Swedish unions and the social democratic party was also strongly opposed to military actions against the Norwegians. The unions were emboldened by a victory against the machine industry in June 1905, where the employers had used a lock-out to drain the cash reserves of the unions but were forced to the negotiation table since the unions had worked together and other unions - including the Norwegians ones - were providing money for the metal workers' union.
The unions and the social democrats threatened a general strike in case of war, and agitated that conscripts should refuse orders to advance into Norway. Members of the youth wing of the social democrats had already been caught (and would eventually be sentenced to prison for sedition for it) distributing leaflets among the Swedish troops to defend the country, but refuse any order to cross the border and invade Norway.
Faced with the absolutely overwhelming Norwegian support for independence, shown both in the parliament and through the plebiscite, the Swedish conservative government realised that even if they could win a military confrontation, they would have no allies in Norway to rule it - 'Vi kan erövra Norge, men vi kan inte styra det.' - (We can conquer Norway, but we cannot rule it).
Finally the negotiations reached a conclusion and the Swedish parliament confirmed the dissolution of the Union on the 26th of October 1905 and King Oscar II abdicated as King of Norway.
So, the bottom line and the answer to your question:
The general populace in Sweden was not as much for Norwegian independence as they were against going to war with Norway to prevent it. The Norwegians were seen as a 'brethren people' and shedding their and your own blood to maintain a union that they had rejected so strongly was seen as folly. While the Swedish army and navy were larger and stronger than its Norwegian counterparts, the terrain favours defence and the level of equipment, technology and training were about equal on both sides. The fact that the Swedish unions had recently, with the benovelent aid of the Norwegian unions, won a victory against a lock-out by the employers had emboldened them and the youth wing of the social democrats were actively agitating among the Swedish conscripts to refuse any order to advance into Norway.
While we can't know how much of a general strike the unions would have been able to organise and whether or not the conscripts would have refused orders or even mutinied if ordered to march into Norway or if the army would have been able to maintain discipline, in the end, the risk seemed real enough for the contemporary conservative politicians, that it combined with the strong and unified Norwegian rejection of the union seemed like a military response was hardly worth it.
Sources:
Varför ledde Norges secession 1905 inte till krig? (Why did Norway's secession 1905 not lead to war?) by Evert Vedung.
Bilden av Branting (the image of Branting), various authors.