War, annexation and frontier settlement.
During the reign of Ivan I (1331-1340) the Grand Principality of Muscovy controlled less than 20,000 sq. kilometers and had a population of several hundred thousand subjects. By 1917, Nicholas II ruled over a Russian empire of 125 million subjects and over 22 million sq. kilometers.
It's a really complex story, involving first the overthrow of the centuries long Mongol-Tatar yoke (refusal to pay tribute to the Golden Horde), competition with rival East Slavic principalities (Novgorod, Suzdal, etc.), leading to the proclamation of a Russian Tsardom by Ivan IV (Иван Грозный, or 'Ivan the Terrible', although this isn't the exact sense of the Russian) in 1547.
So in 1547 you have this new Russian Tsardom in an incredibly unsteady position. To the West and South-West you have Poland, a traditional rival of Muscovy and possessing a much stronger military. To the east, you have a far more complex situation: the Golden Horde, a nomadic empire drawing lineage from Chinggis Khan (Ghengis Khan) and once the supreme power of the Eurasian steppe, had dissolved into a shifting series of semi-nomadic political groups that claimed legitimacy from this Chinggisid Dynasty - and they weren't all that happy about the idea of Muscovy consolidating the lands of the Rus'. Perhaps the strongest of these groups was the Crimean Khanate to the south, which not only was much more powerful than the Russian Tsardom in the 16th century, but also enjoyed favor with the Ottoman Empire, which had a huge military reach and massive amounts of legitimacy in this area of the world in this period. And, on top of all of this, it must be mentioned that for Ivan IV to declare himself a Tsar was an incredibly ballsy move - derived from 'Caesar', this basically signified that Russia believed itself rightful heirs to the fallen Byzantine Empire, which itself links back to ancient Rome (hence Moscow = 3rd Rome).
So, what happens in the following centuries is a series of well-timed wars and skillful diplomacy. Russia was incredibly adept at playing stronger powers against each other, forcing the Crimean Tatars and Poland into disadvantageous wars and manipulating tribal rivalries in the East to gain stronger and stronger footholds. With tribes weaker than itself, the state enforced the payment of Yasak - an annual amount of goods demanded to demonstrate Russian sovereignty.
Russia quickly asserted itself over larger and larger territories of the former Golden Horde (although this process continued well into the 19th century, and in many ways [see the Caucasus] was still an ongoing project in 1917), and at the same time these lessons learned in incorporating scattered peoples into the Russian Empire was taken to Siberia, where military expeditions (often led by Cossacks, a whole different story in the rise of the Russian Empire) pushed onwards to the Pacific with incredible speed in the late-16th / 17th centuries, driven by the insatiable market for Sable furs. The majority of land gained, and indeed possessed by Russia is East of the Urals, and the extremely low population density of these territories facilitated imperial expansion at perhaps the highest rate in world history. At the same time, you have the state (especially from the 18th century onward) seeking to reap as much as possible from its widespread territories, encouraging resettlement, building frontier outposts / forts, etc etc etc.
As to why it got so big, that's a whole different post. There are a few theories hinted at in what I wrote above (the ambitions of the Third Rome, economic factors, the notion of 'Internal Colonialism' coined by Etkind), but I'm not ready to get into such a complex question.
This was just an extremely fast post, and if you're interested in the expansion of the Russian Empire, I have to recommend:
Khodarkovsky, Michael. Russia’s Steppe Frontier: The Making of a Colonial Empire, 1500-1800. Bloomington, IN: Indiana University Press,
and
Slezkine, Yuri. Arctic Mirrors: Russia and the Small People of the North. Ithaca, NY: Cornell University Press, 1994.
Both are classics on the politics and cultural exchanges of Russia's frontier expansion.
–/u/archaeontologist
original thread: How did pre-soviet Russia get so big?