Why did the fighting take place at fort Sumter?

Answer 1

easy. The north had control over the fort but as supplies ran low Lincoln sent a shipment of new supplies for them. He notified the governer of South Carolina that the ship was only for supplies not reinforcements but the South were angered and said that to them supplies were the same as reinforcements. Hence why the South fired on the fort.

Answer 2

Sumter was a US Army garrison on a tiny island in Charleston harbour.

To the Confederates, this was part of their own state of South Carolina, and they wanted to assert sovereignty over it. Lincoln did not recognise the Confederacy as a sovereign nation, and wanted to defend it.

When it proved too vulnerable, he evacuated it, and called for new volunteer troops. This was taken as an act of war.

(There was no formal declaration of hostilities - also because of non-recognition of the Confederacy - as there was also no peace-treaty at the end.)