The Russian Campaign was a huge factor in bringing down Napoleon for two main reasons. The first, less impactful reason was that he lost an absurd amount of horses. He had difficulty reconstituting his cavalry after the campaign, as well as needing horses for supply trains, which played a crucial role in so many of his early victories. The second was the realignment of his "allies" away from France and back towards Russia and Britain.

Consider some of the powers allied with Napoleon on the eve of his invasion of Russia:

The Duchy of Warsaw (but not the Kingdom of Poland, as Napoleon refused to reconstitute the Kingdom, considered by some a strategic error) The Confederation of the Rhine Austria Prussia

Now consider some of the opponents at the Battle of Leipzig in 1813:

Russia Austria Prussia Saxony and Wurttemburg (members of the Confederation of the Rhine who defected during the course of the battle)

The invasion itself resulted in the deaths of most of the Grande Armee, but more importantly it broke down the fragile alliance he established with Prussia, Austria and other German states after previous victorious campaigns. Napoleon humiliated the Prussians after the campaigns of 1806-1807, after crushing them at Jena and Auerstedt, and ultimately led to the Treaty of Tilsit. Prussia lost about half of its territory and population, and crucially their army was limited to no more than 42,000 men. Austria was likewise defeated in the war of the Fifth Coalition in 1809, resulting in the treaty of Schonbrunn. This campaign was of note because Napoleon's German allies bore a large chunk of the fighting responsibility.

The period between Tilsit and Schonbrunn was probably the zenith of Napoleon's empire. This was before the Peninsular campaign had ground down a large portion of his army and demanded ever more resources. Prussia was nominally aligned with France, Napoleon and Tsar Alexander were on friendly terms. Napoleon would, in 1810, choose to marry Marie Louise, the daughter of Habsburg Emperor Francis II, strengthening this tenuous bond after Schonbrunn. The Bavarians and other German states were making real contributions to the war efforts.

The Russian campaign broke almost all of these bonds. The Austrian forces invading Russia sat largely idle after a few minor engagements. The Prussian army, again limited in size by treaty terms, switched allegiance after the Convention of Tauroggen (this was largely facilitated by Carl von Clausewitz, of vom krieg fame). This switch was formalized by the Treaty of Kalisz in early 1813. Many Prussian generals had previously joined the Russian army rather than serve in the Prussian army for Napoleon, so the seeds were sown here already.

After the French army was in tatters and in retreat, opportunists seized the moment and realigned against Napoleon. Russia and Prussia saw the Duchy of Warsaw as vulnerable to annexation. Sweden had its sights set on Norway and enlisted the aide of Britain. The general tide had turned from an advance eastward to an advance westward. Austria, having stronger ties through the marriage of Napoleon and Marie Louise, eventually declared war on France due to diplomatic/territorial disputes with France over the confederation of the Rhine. Finally, Napoleon lost crucial German allies mid-battle at Leipzig.

This is a bit of a winding answer, but basically the events of the 15 or so years leading up to the invasion of Russia saw napoleon flexing his military muscle, and largely bullying the continent into alliance with him. The issue of course is that the other great powers had their own territorial ideas. When they sensed a reversal of fortune for the French, they struck. Russia was on a war footing, and could more rapidly mobilize troops than at any point in previous coalition wars. Austria had a capable commander in Schwartzenberg, and had also began the sort of whole-nation mobilization necessary to combat the French.

Whereas early battles such as Marengo or even Austerlitz saw anywhere from 30,000 to 80,000 men in the field for either side, the sixth coalition wars saw huge increases in army sizes and the amount of field pieces in use. The allies could put a quarter million to a half million in the field across multiple positions. While the French could hastily summon armies of near equal size, they were not of the same quality as the army that marched into Russia, and as previously stated they lacked the horses to build a solid cavalry component, or to draw their wagons.

The Peninsular War played a large part in draining the French Empire of vital resources and manpower. This cannot be overlooked, but the destruction of the Grande Armee opened the floodgates for the opportunists across the continent.