It's a combination of things. In their protests, secessionists accuse the government of the nation and governments of the free states this sort of offense:

Failure to suppress abolitionist groups.

The free states permitted antislavery political organization within their bounds. Those organizations then petitioned the government (as they had a right to do under the First Amendment) for things like banning the interstate slave trade or abolishing slavery in the District of Columbia. They also used the US mail to distribute antislavery literature in the South, though policy dating back to the Jackson years permitted local postmasters to censor said mail.

Personal Liberty Laws

Closely connected to the previous, this has to do with fugitive slaves. Should a slave steal themselves from their rightful owner and go off to a free jurisdiction, slaveholders had a constitutional right to recapture and regain that slave. It's codified in the Fugitive Slave Clause and further clarified in the Fugitive Slave Acts of 1793 and 1850. The latter is the one usually referred to as the Fugitive Slave Act without qualifiers.

Free states did not love these laws. Even if you didn't care about slavery, you might object to the spectacle of potentially free black people just being carried off. So over time they enact personal liberty laws, which they had a right to do under the Supreme Court decision in Prigg v. Pennsylvania. The ruling there was that enslavers had the right to retake their property, and the feds couldn't stand in their way, but the states could. States pass laws requiring forbidding state officials from helping in the rendition of fugitives, denying slave catchers the use of their jails, requiring jury trials for the accused, etc.

Overruling these laws is a major impetus for the second Fugitive Slave Act, which is one of the most radical laws passed before the Civil War. It criminalizes failure to aid in the recapture and rendition of fugitives, establishes a kangaroo court system to rule on these renditions ($5 to a commissioner who rules against rendition, $10 if he rules in favor), and sets up an ersatz draft of any able-bodied northerner around for slave catching. The way this worked was that a black person might run past you on the street. The people chasing him could deputize you on the spot to grab the accused fugitive and you would face hefty legal penalties for refusing. This was all wildly unpopular in the North, which responds with some high-profile rescues of fugitives. Some of these get bloody, particularly the one at Christiana, PA (enslaver killed) and most famously the Anthony Burns case in 1854 where a deputy federal marshal was killed.

Probably more renditions went by smoothly than had spectacular resistance, but it's enough that the old system of informal kidnapping probably claimed more people than the official channels.

The existence of the Republican Party

The Republicans of 1860-1 are a party without any significant support in the South. Their platform is avowedly anti-southern, because to be antislavery is to be anti-southern. Southerners believe that the GOP either has no interest in building a party among them (which isn't true) or will use that party to launch (intentionally or not) a slave insurrection. At the very least, having such a party in the slave states means that slavery will be more openly debated and that is already seen as an existential peril to the white South. An actual attempt at abolition would be catastrophic.

Since slavery is the essence and lifeblood of the South, from which Southern distinctiveness flows and upon which all Southern civilization depends, the GOP is not just an opposition party. The South handled one of those (the Whigs) just fine for a couple decades prior, but the Whigs in the South were a proslavery party supported by major planters. They did not challenge the regime in the way that the GOP did. By endorsing the Republicans and making one of them president, the North had fundamentally broken faith with the Southern understanding of the Union. Specifically, it's a Union between free and slave polities wherein slavery receives extra protection and enjoys the special patronage of the government. In other words, it's a Southern-dominated union which will defend slavery and tolerates free states in limited, qualified ways. The Republicans were not invested in that order, but rather a repudiation of it.

Needless to say, these compacts are all couched in a heads-I-win, tails-you-lose framing. The white North decided over the course of the 1850s that they had had just about enough of all that and rejected unparalleled advances in Southern domination over them (the fugitive slave act included). However, they acted through the normal, agreed-upon machinery of government to do that. Nobody accused Lincoln of staging a coup. The whole point was that the Union had come to permit this, therefore the Union had become a horror to escape.

Sources

Fehrenbacher's The Slaveholding Republic goes through the fugitive slave issue in detail, among many other ways that the South controlled the nation.

McPherson's Battle Cry of Freedom will give you the bare bones. A late antebellum survey will do better. I recommend Potter's The Impending Crisis. If you want something more contemporary, Levine's Half Slave and Half Free covers the same ground with a more social history focus. Varon's Disunion! takes you right up from the Constitution, but stops in 1859. Looking exclusively at the South and getting into the longer reads, Freehling's Road to Disunion (two volumes) covers the same ground as Varon and then some.