If we think of the territory of the Kievan Rus as the "original" Russian territory, then how and why did they expand into Siberia and the Far East? Under the Mongol and Tatar Yokes they were still limited to Europe, if I remember correctly, so how did they move East?
Without being too obnoxiously self-referential, anyone who has seen my answers in this sub concerning the growth and contraction of Mother Rus' during various periods of time knows that I am a big fan of great maps, so please allow me to start this answer with the provision of such a map.
Once you see that green blob grow past the Urals, it's important to point out that any modern map claiming to show the 'borders of Russia' beyond the Ural Mountains (which represent among the last significant natural landmarks in Siberia until you get to at least Lake Baikal, if not the Pacific Ocean) at some historical point in time is built upon a substantial amount of estimation. These lands, even after their conquest, were mostly inhabited by steppe and nomadic peoples and so to claim that anyone-- let alone the cartographers and surveyors of the Age of Exploration-- were capable of mandating or identifying any kind of steadfast borders in an area that had (in its most inhabited regions) a population density of <0.8 people/ km^(2) in the early twentieth century^([)^(1)^(]) (to say nothing of the sixteenth through eighteenth) would be placing an inordinate amount of faith in their abilities.
That said though, the when of your question is answered quite nicely by that map showing that by about 1550, the territory that Russians considered to be Russia had reached up to about modern-day Krasnoyarsk (founded 1628 at 92^(o) E). By 1700 it had reached the Kamchatka Peninsula (the modern-day capital of which, Petropavlosk-Kamchatsky, was founded in 1740 at 158^(o) E) and by about 1760 had reached the effective end of the road in modern-day Anadyr (founded in 1889 at 177^(o) E). The first of those landmarks is right in the center of modern Russia, the second is a huge peninsula in the Pacific, and the final is just about as far northeast as one is capable of going on continental Russia. For more human-readable measurements, keep in mind that 1^(o) of latitude is equivalent to 111 km or 69 miles as the crow flies. So a little quick math here, that means the distances we're talking about here are on the order of several thousand kilometer bursts of expansion eastward at various points of time-- for comparison, the United States of America spans about 4,400 kilometers lengthwise. It took staggered groups of Russian explorers about 60 years to get from the Ural Mountains to the Pacific Ocean-- that feat in and of itself should not be discounted, on account of its sheer audacity alone.
What you essentially had was explorers pressing eastward until they could not believe that they had found nothing but Siberian tundra for the last couple years and just calling it a day, setting up some small armed outpost, planting a flag, and claiming that the land was now nominally a part of the expanded Russian Empire.
There was certainly not-insignificant skirmishing with indigenous peoples that occurred from time to time (probably most notably on the Kamchatka Peninsula), but the sheer volume of land made such incidents far less traumatic or commonplace than, say, the more-famous Americas analog. It must have been absolutely infuriating to read about those glory hounds in Spain sailing back from the New World with literally so much silver that they deflated their own economy into ruins while you were freezing to death in -13 C^(o) eating dried fish and spoiled oatmeal. Not that that ever actually happened, but it produces a pretty funny mental image.
Just a little context for the unacquainted reader there to start, but now, onto the why. I hate to give you an overly terse answer here, but the primary reason to keep pressing onward was economic. As I alluded to earlier, this was all occurring against the backdrop of the European Age of Exploration and the ruling classes of Russia had no intention of missing out on the plunder. What Siberia may lack in sandy beaches, habitable climate, and picturesque mountaintops, it makes up for in gold and silver ore, fur-bearing animals, timber, and natural gas (the latter of which wasn't really exploitable until the late eighteenth century of course). In fact, the near-fruitlessness of the Siberian adventurism of Yermak Timofeyevich (1500s), Vasily Poyarkov (1600s), and Vladimir Atlasov (1700s) is part of what inspired Pyotr I the Great to build his 'window on the west' Saint-Petersburg (1703), establish a reliable navy, and shift his focus westward instead of the increasingly diminishing returns provided by the far east^([2]). That's not to say that these explorers never discovered anything of value-- quite the opposite actually. But the problem is that they didn't really have the technology (or indigenous people to enslave à la Hernán Cortés) to maximize their yield or:
This begins to get us to the how of the whole thing-- by the eighteenth century, the exploration of the now-Russian Far East was increasingly done by ships^([3]). Earlier, these land-based groups consisted of anywhere between fifteen and a few hundred men (and women, who were almost assuredly brought along in the capacity of sexual slaves-- or concubines I guess, if you prefer that terminology) marching eastward without much of an established goal other than 'find stuff to sell and establish military bases.' As unremarkable as it sounds, these expeditions were largely just groups of what amounted to pirates marching off into the Siberian tundra and coming back several years later with booty, or, you know, not coming back because they died of exposure, betrayal at the hands of their cohorts, or some form of armed combat with the locals.
I mentioned encounters with indigenous Siberians earlier, and it's important to point out that although the bulk of these peoples either accepted or hopelessly resisted Russian overlordship, there were several groups of people who managed to severely disrupt these expeditions. The Kamchadal people (from whom the Kamchatka Peninsula derives its name) were notoriously rigid in the defense of their homeland but ultimately suffered the same fate as many of the other peoples between Moscow and the Pacific Ocean-- forced relocation to reservations, insurrection, brutal repression, submission, assimilation^([4]). Why did the Kamchadal put up such a significant and capable fight? There's not really a definitive answer to that question in the historical record as the Itelmen language (which was spoken by the Kamchadal) had no written form until after Atlasov's conquest of their homeland. Furthermore, all but the aforementioned Itelmen dialect of the Kamchadal language were rendered extinct during the Christianization and associated repressions foisted upon the Kamchadal after they capitulated. Perhaps the 15,000-20,000 Kamchadal fought so vehemetly because their material circumstances were so much more dire than those living in the vast expanses of Siberia. The Kamchadal lived on a peninsula which was surrounded by water except for a ~150 km (~100 mi) isthmus connecting it the continent, so they didn't exactly have many options other than to stay put and fight.
Modern-day Kamchadals still have some semblance of cultural heritage such as dialectic language and culture, but for the most part (as with most other indigenous peoples) they have been more or less assimilated into the so-called Russian ethnicity. Think of it like the Cossacks: among people in the know, they are certainly understood to be a distinct ethnic group, but to most non-Russian people on the street, they would be identified as Russians.
So that's really that. How did they do it? The super hard way. They just hoofed it until they ran out of steam. Why did they do it? They wanted to get paid and cash in the first time globalization reared its head upon the developing world. I don't say that to discount the tremendous human achievement of basically walking across a fifth of the globe in a matter of two generations, because it most certainly was that as well.
Sources and Further Reading
[3] Bonhomme, Brian; Russian Exploration From Siberia to Space (2012)
[4] Murashko, Olga; A Demographic History of the Kamchadal/Itelmen of Kamchatka Peninsula; Arctic Anthropology (1994)
[2] Sokol, A.E.; Russian Expansion and Exploration in the Pacific; The American Slavic and East European Review (1952)
