You seem to be under the impression that Congress issued a formal declaration of war upon the Confederacy. No such thing occurred; to do so would be to recognize the Confederacy as an independent and legitimate goverment. As far as the Union was concerned, they were bringing antisocial and criminal elements of their country under control.

When Confederate troops fired upon Fort Sumter the issue had been pressed. Lincoln sent out the call for the mustering of a militia of 75,000, as per his powers under the Militia Act of 1795 (Lincoln's summoning can be found on wikipedia).

At this point a military effort to put down the insurrection was in full swing. Most political officials sympathetic to the Confederacy had either left their posts and joined the the new Southern government or had been placed under strict government scrutiny (Maybe even arrested) and did not have the option of anything less than support of the Union.

In the following months Congress and the Senate gave the presidency more and more powers to muster troops, impose order on any unrest, and these legal actions allowed for the transformation of the conflict into something we'd call a war

As /u/ironcrown9 points out, there was no formal declaration of war. That said, it was clear to people in both the Union and Confederacy that the United States was at war.

The Union advocates of fighting were not uniform in their reasons for supporting the war, and the popular reasons for fighting actually shifted over the course of the war.

In the beginning, Lincoln and an almost unanimous contingent of Republicans and Democrats advocated violence on the basis of preserving the Union. Secession, these people agreed, was not a legal right of the states. It was their view that a vocal and powerful minority in the South was perpetuating the secession crisis over a political issue (slavery). The northern states eagerly volunteered troops, exceeding Lincoln's expectations, in the early days of the war in the hope of preserving the integrity of the Union and the Constitution.

There was some plurality, however. Some radical abolitionists actually opposed fighting the South and adopted a "let them leave and take slavery with them" kind of attitude.

As the war continued it's aim for the North changed, though it did not retain it's relative unanimity. Lincoln's issuing of the Emancipation Proclamation, freeing slaves in Union-occupied areas of the South, in 1863 was extremely controversial in the North. It lost Lincoln the support of almost all "War Democrats" (so-called because they had previously set aside party affiliations to support Lincoln in wartime) and many conservative Republicans.

By 1864 and 1865, as the Confederacy weakened and surrender approached, Lincoln declared that the only terms he would accept would be those of total surrender and the end of slavery. Like the proclamation, this infuriated a lot of people who said this was Lincoln's attempt to prolong the war and aggravate the tiring Confederates. The whole nation was growing very tired at this point, and many states were struggling to find new volunteers for the war effort. People predominantly wanted Union and peace, but Lincoln, and moderate and radical republicans, wanted an end to slavery.

Lincoln masterfully navigated that climate and with the passage of the 13th amendment was able to ensure slavery's legal defeat and thus the war could end without fear of slavery somehow surviving in the South.

If you have more questions or want some quotes/cited sources I can add them in the morning. All of this is coming from a Lincoln-centric perspective and is based on the chapters on the Civil War in David Donald's biography "Lincoln."