How did the Siege of Vicksburg affect the South?

Answer 1

Disastrously.

They lost their last garrison on the Mississippi, and the valuable use of the great waterway was totally denied to them - with their units to the West of the river cut-off from the rest.

It also released Grant's army to join up with the Army of the Cumberland at Chattanooga, as well as giving Grant enough credibility that he would soon become General-in-Chief of all the Union armies.

Answer 2

the union took over the Mississippi river and the city of Vicksburg
You need to be more specific.

Answer 3

The states west of the Mississippi were completely isolated. The Mississippi Valley states ceased to be a threat, and yhe Deep South states lay open to invasion.