Below I have copy-pasted my answer to a similar question, with some revisions.

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TL;DR: Many secessionists understood that Lincoln and the Republicans in the incoming 1861 Congress likely wouldn't be able to get their agenda enacted. But his election represented, to them, a turning point in federal politics. Due to the population discrepancy, the North had now proved they could win the White House and gain Congressional majorities without any Southern support at all. That meant, compromise and concessions from the North would soon be unnecessary, and slavery was certainly going to be an eventual victim. Secession was important in the aftermath of Lincoln's election because there was likely to be no future point when the South would be as united as they then were in support of slavery, and against the Republicans and abolition. It was "now or never". If they did not take a stand in support of slavery at the present moment, Southern politics were sure to soon fracture on the slavery issue, and it would be doomed. Secessionists were willing to go to war to prevent that from happening.

LONG ANSWER:

Good question, and it was one that Northern politicians and moderate Southern politicians from slave states were wondering about out loud at the time, too. Put it into context: Even before the Civil War and the Secession Crisis, there had already been related crises. The most notable is the Nullification Crisis of 1832-33, during which some South Carolina politicians had threatened secession by a "state's rights" or "Compact Theory" interpretation of the Constitution, a view that had never had much support in the federal judiciary (almost certainly not enough to have South Carolina's argument on behalf of nullification upheld by the federal courts). This culminated in the South Carolina General Assembly passing an "Ordinance of Nullification" in November 1832, signed by the governor in February 1833, that ended with a commitment to secede from the Union if South Carolina did not get their way.

Violence was averted by a Congressional compromise, though not before President Andrew Jackson issued a proclamation declaring secession to be illegal. There was a famous Senate debate at the time, too, called the Webster-Hayne debate between Daniel Webster of Massachusetts and Robert Hayne of South Carolina, highlights of which were published in newspapers nationwide. Webster was roundly considered victorious in his defense of the Constitution that it rejected nullification or secession as a legal constitutional remedy to federal-state disagreement.

And at the time, many of South Carolinas's fellow slave states rejected the argument that a state had the right to secede, let alone that it was justified in the 1832-33 circumstances. But over the next couple of decades, the pro-slavery, pro-state's rights wing of Southern politics became more vocal and more extreme, and began to find supporters in almost all the Southern statehouses, particularly in the Deep South.

Another flash point had occurred with the passage of the Compromise of 1850, after which there was a Nashville Convention of representatives from slave states, some of whom advocated for secession in response to the Compromise, though this was still a minority view. Nevertheless, Southerners in Congress lobbied for a more favorable compromise and further concessions by the North, to replace what had been passed by Congress in 1850.

Through the presidencies of Franklin Pierce and James Buchanan, the situation over slavery deteriorated even further, especially due to the fallout from the Kansas-Nebraska Act that led to the proxy war known as Bleeding Kansas.

In the lead-up to the 1860 Presidential election, with the North having organized a party explicitly around the issues of repealing the Kansas-Nebraska Act, of stopping the spread of slavery to new states out West, of repealing the Fugitive Slave Act of 1850, and of appointing abolitionist judges to the federal courts, there were calls by Southern politicians during the campaign that the election of Abraham Lincoln would be tantamount to an act of hostility against the South that could only be resolved by secession.

Not every politician from the slave states saw it that way, however. During the campaign, in August 1860, U.S. Senator John Crittenden of Kentucky gave a speech on the floor of the Senate saying that, even with a Lincoln victory, however alarming that may be to the South, it should not be a cause for disunion. Slavery could survive a Lincoln presidency, and the best course of action was to fight the constitutional way, through the congressional process, against any anti-slavery measures that Lincoln and the Republican Party proposed. Oppose the Republican platform in Congress, then vote Lincoln out of office. Secession was a drastic, and unwarranted act:

"The mere fact of the election of Mr. Lincoln would be a great calamity, though it should not create resistance to the Government. Personally, he is very probably upright, honest and worthy. He married a Kentucky lady, and is a Kentuckian himself. But, politically, he is the agent and subject of the party which brought him into political existence. As the Republicans' President, he would be at least a terror to the South. There is a very considerable Southern sentiment which apprehends much mischief from their success. A feeling of uneasiness and insecurity would pervade [the South].

"But, whoever be elected, [the winner] should be sustained. No State, or set of States, should start up and rebel, and resist by force of arms a president of the United States elected by the people of the United States. No minority should act the dictator unless they are ready for revolution and anarchy. If our President misbehaves, let us call him to account in a legitimate way according to the constitutional forms of our Republican Government, and displace him at the constitutional time."

About six weeks later, on September 22, 1860, Gov. Sam Houston of Texas gave a similar speech in support of slavery but against secession. After giving many reasons why secession would be worse for upholding slavery than would continuance in the Union, he said that Lincoln's election by itself was not reason enough to secede:

"But if, through division in the ranks of those opposed to Mr. Lincoln, he should be elected, we have no excuse for dissolving the Union. The Union is worth more than Mr. Lincoln, and if the battle is to be fought for the Constitution, let us fight it in the Union and for the sake of the Union. With a majority of the people in favor of the Constitution, shall we desert the Government and leave it in the hands of the minority? A new obligation will be imposed upon us, to guard the Constitution and to see that no infraction of it is attempted or permitted. If Mr. Lincoln administers the Government in accordance with the Constitution, our rights must be respected. If he does not, the Constitution has provided a remedy.

"No tyrant or usurper can ever invade our rights so long as we are united. Let Mr. Lincoln attempt it, and his party will scatter like chaff before the storm of popular indignation which will burst forth from one end of the country to the other. Secession or revolution will not be justified until legal and constitutional means of redress have been tried, and I can not believe that the time will ever come when these will prove inadequate."

Of course, Lincoln was elected, and within days, South Carolina was passing legislation to hold a secession convention, and other states had followed suit by the end of November. South Carolina was the first to secede on December 20. Six more states seceded by February 1 of the following year.

There was a lot of negotiation of "compromise" back and forth during the Secession Winter of 1860-61, essentially to make further concessions to the South to prevent secession. But Republican politicians, and many in the North otherwise, already felt they had given way more concessions than they should have under the Compromise of 1850, and the Kansas-Nebraska Act, and Dred Scott, and more, so compromise didn't get very far. Though it did get far enough that the "Corwin Amendment" to the Constitution got its 2/3 approval in both houses of Congress in the last days of the Buchanan administration. However, it was a rather modest concession, basically saying that the federal government couldn't interfere with slavery where it already existed, and making a concession that the feds could never abolish slavery in those states in the future against their will, even if they had the votes to do so. But this didn't do anything to stem the tide of secession.

In the days before South Carolina seceded on December 20, 1860, there was debate on the floor of the Senate over an earlier, failed, compromise (the Crittenden Compromise, in fact, proposed by the aforementioned senator from Kentucky). During the debate, Sen. Andrew Johnson of Tennessee gave a speech that echoed the sentiments of both Crittenden and Houston. Sorry it's a bit long but it is all worth reading (emphasis mine, which is the section most relevant to your question):