The tensions between North and South had been growing since long before Lincoln was elected. While it is true than many in the South believed that Lincoln supported the forced suppression of slavery, his election as a Republican president was simply the trigger for secession.
The story of John Brown's raid on Harper's Ferry in Virginia in October 1859 explains a great deal. Brown had planned to instigate a major slave rebellion in the South, but the raid was poorly planned and ill equipped (less than 20 men without adequate rations). Although the raid was doomed from the outset (he and his men were captured within 2 days), the response from many in the North was widespread admiration.
Brown was hanged for his actions in the raid, but came to be seen as a martyr by many in the North, including the popular poet, Ralph Waldo Emerson. This just fuelled the flames of outrage in the South.
Although the Republican party condemned Brown and the raid on Harper's Ferry, many individuals within that party did not. Again, this caused further outrage in the South. Several Southern politicians blamed the Republican Party for the attack and (falsely) claimed that that Abraham Lincoln supported Brown's intentions. Fake news is not a new phenomenon! As a result, the idea of Abraham Lincoln as President became intolerable to many in the south.
Both sides were becoming more and more polarised. Moderate voices on both sides were silenced (or perhaps simply not reported - moderate opinion rarely sells newspapers!)
Some of the key milestones on the road that led to the secession of the Southern states are discussed on this site, and are well worth reading.
For Abraham Lincoln's opinions on the subject of secession, this site, maintained by the National Park Service is also worth a read.
I'd also recommend watching the first episode of Ken Burns' 1990 documentary series The Civil War. Actually, I'd recommend watching the whole series. In my opinion, one of the best television documentaries ever made.