The Vicksburg campaign was a prolonged siege that could actually be broken down into several different phases, both within Grant's command of the operation, and prior to that. Grant's campaign alone lasted almost 4 months, and was the continuation of an extended operation to take the Confederate city/river stronghold. The fall of Vicksburg on July 4, 1863 was the culmination of a brilliant overland campaign that Grant conducted that was the precursor to Sherman's "March to the Sea." Grant abandoned his supply lines when he marched south of the city (across the opposite bank of the Mississippi), crossed the river, then cut Vicksburg off from Jackson, essentially isolating the city and sealing its fate. People thought Grant was crazy to march off without his Army umbilical, but he correctly predicted that his troops could carry what vital provisions they needed, and could live off the land on the way to their objective(s). Again, this strategy was nothing short of a tactical breakthrough, and was part of the blueprint that put the final nails in the coffin of the Confederacy.

So, strictly from a tactical perspective, Grant's campaign was revelatory, and a HUGE influence on the Union's prosecution of the war going forward. As far as why it was "a turning point," the main reason was that the fall of Vicksburg meant that the Union had complete control over the Mississippi River. This meant that the Union could use this massive waterway as a highway to move men and materiel across vast portions of the "Western" theater of operations. Imagine if in a modern military campaign, one side lost the ability to supply their forces by air, and the other side gained it. The impact was no less significant. Having a major waterway connecting literally a thousand miles of territory at your disposal was a massive advantage in the 19th century. Both Jefferson Davis and Abraham Lincoln made comments about the significance of Vicksburg, and their words bear consideration. Lincoln remarked after the city's fall that "the Father of Waters again goes unvexed to the sea." Davis, prior to this, said of Vicksburg, "Vicksburg is the nail head that holds the South's two halves together." Davis understood that losing the city meant that the western half of the Confederacy was essentially sealed off from the east. It was a literal example of dividing and conquering when the Union took it.

The other big reason (albeit coincidental in nature) was the fact that Vicksburg fell the day after Lee got turned back at Gettysburg. Thus, in the span of just two days, the Confederacy suffered two MASSIVE defeats. It was a devastating one-two punch that the Confederacy never really recovered from.

[Sources: Doris Kearns Goodwin, 'Team of Rivals'; Bruce Catton, 'Grant Takes Command'; James McPherson, 'Battle Cry of Freedom']