The Quakers are an interesting community in many ways. As you noted, they're famous for their pacifism. Analyzing the reasons behind why a man or a community fought in a war is a complicated subject, and of course the reasons I'm going to give are general ones, but the answer to your question lays in the Quakers anti-slavery position, and how ultimately they came to believe that fighting for the Union and for emancipation superseded their commitment to pacifism.

Just to give some context, Quakers had engaged in anti-slavery work for a long time. Even before the American Revolution, some Quakers had spoke up against slavery, condemning it as contrary to Christian principles. They would later form the Pennsylvania Anti-Slavery Society, which presented a petition to the first Congress, requesting the abolition of slavery. In general, Quakers firmly believed in the principle of the "guiding light" that naturally conducts every person to do the right thing. As such, a slaveholder would eventually come to his senses and realize that slavery was not Christian, and would then emancipate his slaves. For this reason, Quakers supported moderation and Democratic means, such as petitions or sending priests to Southern states to make slaveholders see the error of their ways.

The Quaker position thus contrasted with the political abolitionism of Salmon P. Chase, or the militant abolitionism of William Lloyd Garrison. In the case of the former, Chase developed a theory that held that slavery in the territories, or even slavery itself, was unconstitutional, and that the government had to ensure freedom wherever it had power ("Freedom National"). In the case of the later, slavery was categorized as a monstrous sin that should be eradicated without regard for Constitutional issues. The Quaker view contrasted with Chase's because political abolitionism did not give much emphasis to the moral element, and although this made it more appealing and thus more successful, Quakers always focused on the fact that slavery was morally wrong. On the other hand, they agreed with the Garrisonians, since both groups always put morality to the forefront. The problem is that Garrison was very militant in his approach, condemning not only slavery but slaveholders as well. In the Quakers' eyes, slaveholders did not know better; in Garrison's, they were "man-stealers" who should be punished for their sins.

A shift within the Quaker community as well as within most pacifist anti-slavery people took place during the 1850's, when outrages such as the Fugitive Slave Act or the Dred Scott decision gave credence to the idea of the Slave Power, a conspiracy to protect and expand slavery. The Fugitive Slave Act is general caused great outrage because it brought slave-catchers to the North, where in many occasions dramatic events took place as abolitionist sought to protect former slaves. One of my favorite stories is that of Reverend Theodore Parker (not a Quaker), who told President Fillmore that he'd rather starve than give up any of his parishioners. The Quakers, known for sheltering fugitives and protecting them through legal action, and for their participation in the Underground Railway, also were forced to choose between their commitment to nonviolence, or their pledge to defend the runaways. They chose the latter.

On September 11, 1851, a slaveholder approached the little Pennsylvania town of Christiana, where two of his slaves had escaped two years earlier. Armed Black men and White Quakers vowed to resist, and when the slaveholder refused to leave peacefully, shooting broke out. The slaveholder died in this "Battle of Christiana", which many Southerners regarded as the natural result of "fanaticism." Vowing to their demands, the government tried to indict the Quakers and their black allies for treason, but the case degenerated into farce and charges were ultimately dropped. The Christiana event, more than anything, shows that when pushed to choose between pacifism and abolition, Quakers often chose the latter. For them, slavery was a sin, and to aid in the capture of a fugitive slave would be to commit a sin themselves. If the Slavocrats used violence to upheld a sin, they were justified in using violence to prevent a sin.

This turn from pacifism towards action, even violent action, was mirrored by many who felt outraged by the violent acts of the South. Seeing events such as the Fugitive Slave Act or the debacle over Kansas as Southern violence for slavery, former pacifists were now more willing to fight fire with fire. Frederick Douglass, for example, had said that he would not abolish slavery if it cost a single drop of blood. But now he believed that "The only way to make the fugitive slave law a dead letter is to make half a dozen or more dead kidnappers."

Quakers wouldn't take action themselves, of course. They wouldn't do as John Brown did, and organize a raid with the explicit purpose of starting a bloody slave insurrection. But when the hour of truth came and the South seceded, Quakers were imbued by the same spirit of nationalism as the great majority of Northerners. But they also saw the war as an anti-slavery crusade, a necessary war to protect against the violent aims of the rebels, who wished to extent slavery and destroy the Union. If violence was needed to protect the Union, so be it.

And so, Quakers overcame their pacifism and enlisted into the Army to fight for the Union. Once in the Army, it seems that the Christian faith of many Quakers was just reinforced by the experience of war. They, and other Christians, acquired a sense of fatalism, believing that God Himself had created the war to purify the nation's sins, and that whether they fell in battle or not was in His hands. "I know He watches over all," said a New Jersey Quaker for example. Others were motivated by comradeship and loyalty to their unit. When faced with the reproach of his wife and sister, and the threats of expulsion from his community, a Quaker soldiers said that he couldn't leave "the men who stood manfuly by me in the hours of dainger through which I have passed."

Abolitionism was a strong motivation for many Quakers. Some came to the conclusion that God had caused the war to end the sin of slavery, and as such fighting was right. A Quaker in the Navy wrote that "slavery is such a horrible blot on civilization, that I am convinced that the war will exterminate it and its supporters, and that it was brought about for that purpose by God." Two Quaker brothers enlisted in the Army, despite the disapproval of their mother, after the Emancipation Proclamation had turned into a war for Liberty. "Thee knows how I have always felt about war," one of them wrote to their mother, "still I never regret that I am a participant in the struggle . . . [to] advance the cause of universal freedom. . . . Remember, Mother, that while we strike strong telling blows for liberty ... we need not hesitate as to the means employed." Even after he fell in battle, the other brother still felt "glad that I was a soldier and have done my part for the cause of universal liberty." Another fighting Quaker condemned the people who stayed home and didn't fight, calling them the "puny cravens at home, whose fears make them tremble at shadows," though he was writing of all people in General, not just his fellow Quakers.

Not all Quakers could overcome their pacifism like this. Many of the members of the communities to which the fighting Quakers I quoted came from, even relatives, condemned them for abandoning pacifism and fighting directly. This is not to say that Quakers turned towards conservatism as a result of the war, not by a long shot. For example, they sent a message to Lincoln asking him “not to allow the present golden opportunity to pass without decreeing the entire abolition of slavery throughout the land.” The Quakers continued to work against slavery. Many were able to overcome their pacifism only after the Emancipation Proclamation made abolition an explicit goal. But before that, as I already mentioned, Quakers enlisted in the Army because they believed that the war could ultimately lead to abolition.

So, to answer the first part of your question: There were many Quakers who saw the Civil War as another example of Southerners violently asserting a sinful institution, and a large number of Quakers were motivated by patriotism and defense of the Union, since they had come to believe that they were justified in using violence to defend against a sin. Yet, they also realized that slavery could perish as a result of the war, and worked arduously to ensure this. Some believed that God Himself had caused to war to end slavery and punish the nation for its sins. A lot of Quakers were able to overcome their pacifism only when the war's purpose shifted and became a war for liberty as well. However, there still remained many Quakers who clung to pacifism. As for motivation, some Quakers were motivated by religious fervor; others, by comradeship. But as a whole, Quakers remained overwhelmingly anti-slavery, and the war only hardened their convictions and pushed many to forget pacifism and fight directly.