How did the election of 1860 increase sectional tensions?

Because the newly-elected Lincoln would not agree to any extension of slavery. So the final attempt at compromise failed.

Prior to the outbreak of the Civil War, the election of 1860 increased sectional tensions of the United States by lifting pro-union, anti-slavery statesman, Abraham Lincoln into the Presidency. In reaction, the Southern States protested in many ways, but in no more dramatic way than, one by one, seceding.