Lincoln's name wasn't on the ballot in most of the states you mention, and South Carolina didn't hold presidential elections at all.
And that matters how? The premise is still the same.
Shoulf I have phrased it:
Imagine a President getting elected who isn't even on the ballot for you to vote against in your region ...
Or perhaps:
Imagine a President getting elected who can't even be bothered to put his name forward on the ballot in your region ...
The country was torn up over the issue of slavery and here a Presidential candidate runs for the office without even so much as bothering to put his name on the ballot in the most affected area of the country. What gall!
It would be as if the overriding issue of the day were water rights and a President won an election when he did not put his name on the ballot in the western states. Imagine some liberal effete easterner not even connected to the issue winning. Here you have a liberal mid-westerner winning the Presidency over the issue of slavery who doesn't come from a slave state and doesn't even bother to put his name on the ballot in the slave states. Actually, that's even more appalling.
You must look at the issue from their point of view. No one in the south today is advocating secession, but they were in 1861. They thought differently then. If you want to understand history you need to understand what motivated them then, not how we would approach the issue today.
It matters because it explains why President Lincoln didn't get a single vote from the southern states. It wasn't a matter of him not wanting to be on the ballot, it was due to the deliberate action of the southern leadership to keep him off the ballot. Who gets placed on ballots on the ballot is a state issue, it still is. Candidates today still need to qualify in every state. Allowing a vote for any presidential candidate in the first place is a state issue, witness South Carolina which didn't hold a presidential election for decades prior to the end of the Civil war. So suggesting that President Lincoln deliberately kept his name off the ballot in the south is the exact opposite of the fact.
We do understand it, from their own words. They were losing power to the increasing number of free states entering the union. By the election of Lincoln, the south had lost the ability to control the government and protect slavery. Slavery was embattled and the south was losing footing. With the election of Lincoln without a single slave holding southern state, they could see the end was near.
Secession was meant to preserve slavery. After 300 years of the south fighting to maintain slavery, it seceded when the tide began to turn. Within the blink of an historic eye, just five years from the south pulling out of the union, slavery was abolished. Clearly only the south managed to maintain the institution of slavery.