From 1789 to 1799, France was embroiled in the French Revolution and much of the government was paralyzed or outright destroyed without competent replacement. The French Republic established in 1799-1804 was equally chaotic, and Napoleon's coronation in 1804 saw the wars started the year before accelerate into a brutal campaign that saw the French army ranging across the breadth of Europe and attempting to administer Germany, Italy, Poland, and huge sections of the Adriatic coast.

It is safe to say that French military attention was heavily divided during this period, and was not focused on the New World at all due to the blockade the British established over French European holdings almost immediately.

On top of this, French forces garrisoning Haiti were minimal, and African slaves and their immediate descendants outnumbered white and mixed residents more than tenfold. For a time, the revolutionaries were supported by Britain and Spain, before recoiling in horror as they realized a black ex-slave republic was likely to succeed.

Add to this the surprisingly canny and effective leadership of African guerrilas and generals, who were able to use the chaos to amass arms and equipment from the colonial powers. Toussaint Louverture began his military career fighting for the Spanish against the French--then for the French Republic themselves against Spain and Britain before taking up the revolutionary mantle. If it were not for the effective and impassioned work of the freed slaves, Haiti would be a memory.

TL;DR: French distraction, British and Spanish intervention, but mostly a huge and highly motivated population with good leadership.