In general, southerners didn't like Lincoln long before the 1860 election. As a member of the Free Soil Party (before it later became the Republican Party) Lincoln argued against expanding slavery into any of the new territories. To them, he was a threat, and represented the worst of northern politicians. That is why in 1860, Lincoln's name did not appear on the ballot in ten southern states. You couldn't vote for him there if you wanted to.
When Lincoln won the election anyway, and with 41% of the vote, the South had had enough. South Carolina was the first to secede in December of 1860, and ten more states would break away between then and March of 1861, leaving Lincoln to take the Presidency of a country that was disintegrating.