The year is 1380. A storm is gathering in Kulikovo. To face off against the innumerable heathen Tatar hordes are the 1.3 million noble knights of Muscovy and her allies, led by Prince Dmitry, whose glorious victory in this battle would have him go down in history forever as Dmitry Donskoy, or in English, Dmitry of the Don. Kulikovo is where the Russian nation has born, where the shattered heathen would never more dare steal from the righteous folk and their lands.

...

If that sounds like a tall tale, you're on point dear reader. The caprices of High Medieval chronicling aside, the rise of Muscovy was not outlined by setpiece battles or daring acts of defiance. Indeed, many attempts from various Russian princes met negative results ranging from inconvenient to disastrous; more often the latter. In such an environment, Muscovy did not rise through acts of heroism, but through patience and... trade. And so, while Kulikovo was not an insignificant battle, we must go back further, before we can understand in what way it mattered, and what came after.

The Rus peoples' trial with the Tatars go further back than Muscovy, which itself was a vassal principality in a then insignificant part of modern Russia, little more than a trading settlement along a tribituary of a tribituary of the Volga. So for such a dark horse canditate to rise, you might be thinking that several things might have had to go right, and you'd be correct.

Muscovy was in luck for three reasons:

The first of these reasons was that the Mongol invasion and the subsequent, almost habitual razing of Rus centers of population and trade meant that the old centers of gravity for Rus culture no longer held their importance. Muscovy itself was also pillaged and set to torch in this period, but, owing to its remote position, and not much wealth and significance to begin with, it emerged relatively unscathed. From here on came the second reason for Muscovy's rise.

Muscovy had a series of wise princes; under the Tatar yoke, a prince had to be wise or he would not last long at all. The most significant one in the build-up to the Battle of Kulikovo, and the one I want to talk about, is Ivan I of Moscow, or as his pals and debtors called him, Ivan Kalita, or in English, Ivan the Moneybag. Now I've talked about Kalita a bit in another question, but to repeat myself, Kalita, after witnessing the execution of three of his predecessors for either insubordination or being victims of court intrigue at the hands of Öz Beg Khan, decided the best way to keep his head was to keep it down, avoid any military adventurism, and above all, pay his tribute in time and in full. To achieve this, he had to build a strong economy resilient to the usual adversities an agriculture-taxation based economy would have in the geographical conditions of Russia, and so he prioritized the promotion and expansion of trade and a vibrant city life, one that counted on a large and concentrated population. To achieve this, he purchased captured slaves from his Tatar overlords, invited refugees, and resettled people in his lands. Using these riches he was able to gather through his shrewd management of the economy as well as excellent diplomacy with his overlords (he managed to convince them to allow hereditary succession for Muscovy so that his son could succeed him), he engaged in what's curtly described as loan sharking to peacefully take over the nearby lands of Russian rulers small and larger. His successful diplomacy, willful obedience and economic cunning made him the earliest identifiable figure in the Russian ascendancy, and expansion of Russian soft power through these peaceful means would remain the only successful and viable way for the next 150 years.

Now that last sentence there may be a bit of a spoiler, as Ivan, being Dmitry Donskoy's grandfather, reigned only 18 years before him, so Kulikovo can't have accomplished all that much if, quoting myself, nothing much could have been accomplished outside peaceful means in this time period. But before we get there, I want to get the third reason why Muscovy was lucky out of the way:

The Golden Horde, the friendly neighborhood Mongol Empire remnant, had high levels of infighting. Now, any medieval polity had infighting, but nothing brought a people together better than being subjugated by another, and nothing led to as much infighting as being a martially inclined society with few external opponents but several claimants to the Empire, each of whom could muster massive armies. Kalita's rule saw Muscovy more or less subsume every significant Russian principality with the notable exceptions of Novgorod and her periphery, and the parts of Ukraine and Belarus which were conquered by the Poles and Lithuanians instead (not yet together, these were separate conquests and the two nations wouldn't come together until a hundred years from then on). So, Muscovy had grabbed all there was to grab, was slowly extracting concessions through obedience and timely payment of taxes, and their overlord was essentially having a crisis on every succession. All Muscovy really had to do, at this point, was to bide their time, and wait for the Golden Horde to crumble on its own.

In 1359, the Khan was killed, and the Golden Horde erupted into civil war. The Khan's son-in-law, a Warlord known as Mamai, played the role of a kingmaker in this time, and consolidated his control over a part of the horde. After a few Khans, he briefly installed his own wife as the Khanum, whom was replaced later with another puppet. So it is fair to say that he was the ruler of the Golden Horde in this period, though his hold would have been tenuous given that he was not a descendant of Genghis Khan, but simply someone who married one of his descendants. After Mamai consolidated enough power, he decided to punish Dmitry, who had refused orders from him to give up the throne of Vladimir. In 1378, this culminated in the Battle of Vozha River, which Dmitry won. Gathering a larger force, it would be in 1380 that the Battle of Kulikovo, the one I led with, would be fought. This was a larger battle with nevertheless indeterminate numbers; the million and change I led with is from a medieval source that is more dramatic than historical (as sources of the period wont to be), and the more likely number is far less than 100.000 on the Muscovite side. Nevertheless, Kulikovo was won, and it was significant; it was the first major victory of Moscow and is considered to have hinted the Tatar decline.

Unfortunately for Dmitry, the decline hadn't come fast enough. The defeat of Mamai in Muscovite hands made Tokhtamysh the undisputed ruler of the Golden Horde, who, after defeating Mamai handily in a battle, moved on to sack Moscow in 1382, and force Dmitry to once again give his oath of loyalty as a vassal. Does this mean all of this had been for nothing? Of course not! Dmitry's son, Vasily I, would be the first Grand Duke of Muscovy to accede to the throne without permission from the Golden Horde, and the Tatar hold would continue to decline over the next 100 years up to the Stand on the Ugra River, which, if such things are reasonable to have in the context of history, was the point at which it can be said that Muscovy was practically no longer under Tatar suzerainty. Between Kulikovo and the Stand (called as such because two armies met but retreated without a fight), Moscow was of course burned by Khans more than once, so it's difficult to say that things moved along in an entirely linear fashion. But the progress was there, and, though it took time, Muscovy went on to emerge as the dominant Russian principality.

To summarize, Muscovy had princes that slowly (and more rapidly in the case of Kalita) expanded her power and significance, while the Mongols gradually declined through reasons that are outside the scope of this answer. As such, the ascendance of Muscovy went from a historical fluke to an inevitability.