No, it was absolutely not uninhabited.

I am not an expert on the Russian conquest of Siberia, and I will not claim to be. However, I can speak to the Russian subjugation of the land west of the Urals, which was inhabited (and continues to be inhabited!) by ethnic groups entirely unrelated to the Slavic Russians practicing alien faiths such as paganism and Islam.

In order to understand how Russia came to control this territory, one needs to go back to the time of the Mongol Empire. When the Mongols arrived, the formerly united Kievan Rus had largely disintegrated into smaller principalities that quickly fell to the Mongol forces. Upon the breakup of the Mongol Empire, this meant that the Russian principalities fell under the Golden Horde, which itself would later break into several parts - the Khanates of Kazan, Astrakhan, and Crimea. Crimea is largely irrelevant to Russian expansion eastwards, as it would not be conquered until the 18th century and lies west of the Russian heartland, anyway. The Khanate of Kazan was an entirely different story. From its beginning, Kazan was an unstable state. Its ruling class consisted of Muslim Tatars ruling over a largely pagan peasantry of Uralic peoples such as the Mari, Mordvin, and Udmurts. This made it ripe for the picking by the rising Grand Duchy of Moscow, which had emerged from Mongol rule as the most powerful Russian principality and was rapidly integrating and conquering its Russian neighbors, consolidating the core of what would become the Russian state.

Kazan existed for slightly over a hundred years (1438-1552). In that time, the Muscovites managed to place a puppet on the throne three times - hardly a signifier of a stable state. However, power shifted back and forth from the Muscovites and the Crimeans until finally, in 1552, Russian forces under Ivan the Terrible sacked the city and brought it under Muscovite rule. This was the first major expansion of Moscow into a non-Slavic region, and a major eastward conquest. (It was also what inspired the construction of the iconic St. Basil's Cathedral). Four years later, Astrakhan would also fall to the Russians, and the nomadic Bashkirs would swear their loyalty to the Russian tsar, bringing Russia essentially to the Urals. Of course, actual control of this territory was hard to achieve, and it was a hotbed of rebellion for much of its existence. The Bashkirs revolted multiple times, and Cossack rebellions aided by the region's minorities made full control of the region difficult until the 18th century.

To exert control over the area, the Russians relied on building forts to defend against raiding steppe nomads like the Kazakhs or Kalmyks (or even the supposedly loyal Bashkirs). Some of these would eventually become some of Russia's largest cities - see Saratov, Samara, or Ufa. That being said, much of the region lay outside of formal control, and huge amounts of serfs fled eastward in an attempt to escape and gain freedom. Large-scale Russian colonization would not occur in the area until the 18th century and conversion attempts were similarly delayed (though the Russians hardly even attempted to convert the Muslim Tatars and Bashkirs).

From there, the Russians marched on into Siberia - a topic which is largely outside of my knowledge, but by and large Siberia lacked any actual states like the Khanate of Kazan (aside from the Khanate of Sibir, from which we derive the name Siberia).

One more note - the Kievan Rus are not solely the predecessors to the modern Russian nation, they are also the predecessors to all East Slavs, including Ukrainians and Belarusians. Their "region of control" was the the East Slavic heartland, and the reason it seems so small is mainly because Siberia is very, very, big.

Sources:

"Muscovy's Conquest of Kazan" by Brian L. Davies

The Russian Conquest of Bashkiria, 1552-1740 by Alton S. Donnelly

Taming the Wild Field: Colonization and Empire on the Russian Steppe by Willard Sunderland