How did the split of the Democratic Party affect the outcome of the 1860 election and the Southern states' decision to secede?

The split insured Lincoln's election and that led directly to secession.

The Democratic Party could not agree on a party platform, so the southern branch of the party bolted from the convention. They nominated their own candidate, John C. Breckinridge, who was the sitting Vice-President and a strong pro-slavery politician. The northern branch of the party nominated Stephen A. Douglas, an Illinois Senator who had southern sympathies, but not enough to please the southern half of the party.

This split in the Democratic Party ensured that Republican Abraham Lincoln would win the election. This he did with less than 40% of the popular vote.

Southerners had threatened secession if a Republican was elected president. It's kind of ironic that it was their split of the Democratic Party that insured their own defeat.

Some six weeks after Lincoln was elected, South Carolina seceded. This was a direct result of Lincoln's election. Eventually ten other southern states would also secede.