The "Southern Strategy" was aimed at severing the Southern states (or colonies, as the British still imagined them) from the North. In this way, they would break the essential stalemate that had emerged in the war in the North, one which cost increasing quantities of British lives and treasure without yielding decisive victories. It was premised on the idea, promoted by former British royal governors, that the Southern colonies were full of Loyalists, who would emerge en masse after a British invasion. The British also thought that rebel leaders, especially in South Carolina, would be frightened by the prospect of massive slave uprisings, and would therefore offer little resistance.
The strategy met with initial success after the fall of the city of Charles Town, the largest city in the South, in 1780. But as the British entered the countryside, things became more confused. Many Loyalists did indeed emerge, but most Carolinians seemed reluctant to openly support the British. In fact, one of the aftereffects of the British invasion was a brutal civil war between Loyalists and Whig partisans in the Carolina backcountry that lasted until 1783. The British were constantly harassed by partisan fighters, and their attempts at retribution, which were on several occasions quite brutal, served to galvanize many of their enemies. In any case, the British were not able to put Southern governments into Loyalist hands—there were simply not enough of them to maintain control of South Carolina and North Carolina.
Along with lack of Loyalist support, another reason the so-called Southern Strategy failed was the strategy pursued by the Continental Army under General Nathanael Greene (as well as the ill-advised response to it by British General Cornwallis.) Greene, as Washington had done in the North, refused to commit his forces to an all-out battle, instead drawing Cornwallis deeper into the backcountry and away from his supply lines. A series of tactical defeats caused Cornwallis so many casualties that the British general thought it necessary to end his campaign. This is what led him to Yorktown, where his forces were bottled up by Continental and French forces and denied naval support by the French navy. So, in short, the Southern Strategy failed because of a misunderstanding of the political realities in the Carolinas as well as poor campaigning by British leaders.