Any relationship between slavery and the shelling of Fort Sumter is remote at best. Slavery was, of course, a defining political issue during the first half of the 19th century, as illustrated by the Missouri Compromise and the Compromise of 1850, both of which attempted to maintain stability in an otherwise volatile union.
When Lincoln was elected President, South Carolina seceded from the Union together with several other Southern states. This was because the South believed Lincoln would work to prevent the spread of slavery to the west. Such a move would be the death knell of the "peculiar institution." There was some debate as to whether South Carolina could legally secede from the Union (a question which the war itself purportedly answered); however it did in fact claim to have left the Union. Immediately after its purported secession, the State of South Carolina seized all federal arsenals and post offices within the state. A garrison originally at Fort Moultrie under the command of Major Robert Anderson left that post and fortified itself at Fort Sumter. From there he refused to surrender the fort to the State. Originally, South Carolina had hoped to starve out the troops at the Fort. When President Buchanan sent a ship, the Star of the East to carry supplies to the Fort, the ship was fired upon and was forced to withdraw. Thereafter, President Lincoln wrote to the Governor of South Carolina and informed him that he intended to resupply the fort with food, medicines, etc but no weapons. It is noteworthy that Lincoln wrote to the Governor, not Jefferson Davis, President of the Confederacy; as Lincoln obviously believed that South Carolina had never legally left the Union. It was only after receipt of this message that Citadel Cadets opened fire on the Fort, forcing it to surrender. The attack on Fort Sumter was, of course, the opening salvo in the Civil War.
So, Fort Sumter was the culmination of a long standing debate over slavery which led to the Civil War; but its actual connection is remote.
The institution of slavery helped lead to the shelling of Fort Sumter because it helped to lead to the political conflicts between the North and South.
The institution of slavery helped to make the North and South different from one another. It made their cultures different and it made their economies different. Because the two regions were so different, they started to grow apart and to feel that what was good for one region was bad for the other.
When Abraham Lincoln was elected with no Southern votes, the South felt that the US government was going to be run only for the benefit of the North. This led to South Carolina and other states seceding, which is what caused the shelling of Fort Sumter.