In their Declaration of the Causes of Seceding States, each of the states outlines their own grievances with what they see as Northern refusal to uphold their right to own slaves (for example, that the North isn't complying with the Fugitive Slave Law), that the expansion of the US (creation of new states/territories) are set up to make slavery illegal whereas they want to expand it since it makes slaveholding states a minority in the congress at the time, and that there are powerful abolitionists in Northern government (esp in Lincoln's party) which they suspect are also going into the South to try to spread what they consider anti-slavery propaganda. As a result, the Southern state seceded. They caused the situation as before hand, Lincoln was morally opposed to slavery (though personally racist) but he was willing to maintain slavery to preserve the nation. So the South didn't seceed because the North was ending slavery- it's the other way around.

As for the North, some of them were abolitionists on moral grounds- and this does not mean they weren't also racist themselves. One of the main arguments that Christian abolitionists had against slavery was that it corrupted the souls of the white slaver and trader. Others were separatists who wished to establish a white nation and saw the growing slave population as a threat to that- you have to remember that in that time, many white planters lived in constant fear of slave revolt, the slave population was growing every generation especially in the decades just before the Civil War, and of course there's the fact that slavers often impregnated women slaves leading to a whole hierarchy of mixed race people - all of these things offended people who wanted to establish a white state.

Many others favored abolition because of political reasons- a lot of the aforementioned Declarations were about the balance of free states with slave states in Congress and this meant who had which interests represented more. This had been an on-going problem which had already come to a head during the Missouri Compromise a couple decades before the Civil War. You have to remember that at this time, the US was constantly expanding westward and they had the goal of dominating all of North and South America so anyone who was looking out for their own interests would see increased representation of other interests in Congress as possible competitors, and the Southern economy depended upon slavery while the Northern did not. So some of this is about economic and political competition.

Also the Fugitive Slave Laws that I mentioned above were a real point of contention. Slaves kept running away to the North and the Southern states wanted the Northern states to cooperate in allowing them to be seized and returned. Northern states didn't always want to do this, it increasingly became a problem for them. This even went all the way to the SCOTUS in Prigg v Penn. It's an irony of history because the revisionists now say the South was all about states' rights, but one of the things they were most angry about was that the Northern states retained the right to not assist Southern slave hunters in returning escaped slaves. The SCOTUS rule said the Southerners could come and get their "property" but that the Northern states had the right to refuse them help in doing that. Every single state in the Declarations I linked to above cited the North's lack of cooperation with the Fugitive Slave Law as a main reason. So much for states' rights, lol.