Through similar processes to how the US later colonized North America, and indeed how Europeans in general conquered America.

First, some things of note. As you'd expect, Siberia, meaning basically all of Russia east of the Urals, is almost an entire continent in itself, and incredibly diverse culturally and ethnically. But pre 1582, the majority of it was very sparsely populated, and mostly by hunter-gatherers and reindeer pastoralists with no political organization.

The one major state that the Russians encountered during the expansion was the Khanate of Sibir, one of the last successor states of the Mongol Empire. This state was destroyed by Cossack conquerors over two decades, using mainly one tactical military advantage: Firearms. These were a key element of Russian military success against numerically superior foes all over Siberia.

The other thing is that Siberia was uniquely navigable for a huge continent-sized regions. Most of Siberia is a geographical plateau covered by vast rivers, and the rivers are connected through relative short portages. This made communication and travel remarkably easier than basically anywhere else in the world, and allowed the Russians to travel over big distances faster than other major European powers. If a message was sent with the highest urgency, it could get from Moscow to the Pacific in six weeks. Getting one from Britain to India in the same time would take six months, so there is no doubt that Asia was a lot closer to Russia in this regard.

It is also worth noting that Eurasian diseases had not reached much of Siberia, so through the 16th and 18th centuries, smallpox epidemics were common among natives, further weakening their ability to resist conquest. this came on top of Siberia also being very sparsely populated, much less people than in contemporary North America for comparison.

In short, the Russians had a set of unique geographical circumstances enabling one of the most impressive expansions in human history. They could travel through rivers, use firearms to beat disorganized and disadvantaged resistance, and could build fortresses, ostrogs, at key junctions of the region to secure regional power. Almost every time the Russians encountered coordinated resistance, their expansion was checked. They never expanded into Central Asia before the 19th century, and were also checked in Mongolia, areas where these geographical advantages ceased to exist. The most notable of these defeats happened in the Amur region, where they were beaten in a series of conflicts with Qing China, eventually removing them fully from the region.

The main source of this is Eastward to Empire by Lantzeff and Pierce, and Conquest of a Continent, by Bruce Lincoln.