What destroyed Napoleon's army in Russia in 1812?

Mostly not the cold. Napoleon's army was worn down by hunger, disease and impossibly long supply lines well before winter, and the Russians kept it that way by retreating and preserving their own army. Winter, when it came, finished the job.

Napoleon began the invasion to force the Russian Empire to comply with the continental blockade of the United Kingdom. Tsar Alexander had broken away from that blockade on 31 December 1810, because it strained Russia's economy. As the Russian army kept retreating deeper into Russia, Napoleon had to rely on an increasingly overstretched supply system that could not sustain such a large army.

The one big fight was Borodino, on 7 September 1812, with more than 250,000 troops and at least 70,000 casualties. The Grande Armée took the main positions but failed to destroy the Russian army. About a third of Napoleon's soldiers were killed or wounded, and Russian losses, though heavier, could be replaced, unlike the French army's. Kutuzov then chose not to defend Moscow. Napoleon entered on 14 September, surprised that no delegation came, and within 48 hours three quarters of the city was ashes. He stayed five weeks, left on 19 October, and winter arrived on 6 November with a blizzard, while the army was still in summer clothing.

The popular saying is that "General Winter" beat him. But Minard's map shows that French losses were highest in summer and autumn, because of poor logistics and disease. It is estimated that of 612,000 combatants who entered Russia, only 112,000 returned.