I can give a brief answer to say it was a combination of trade, military, and population pressures.

One of the chief instigators of eastward Russian expansion was the Siberian fur trade, which boosted the Tsardom's income as more of it came under their sway. While earlier agreements and levies from native populations provided the Russians the furs they traded elsewhere, the enterprising Stroganov family was party to direct Russian encroachment and control of fur sources farther and farther East and North, where the pelts were thicker and thus more expensive. Much like later European trade companies elsewhere, Stroganov expansion was supported then coopted by the Tsardom, leading to Russian expansion.

Another reason was competition with the many Turkic states in the region. The Russians had the remnants of the Golden Horde, particularly the Crimean Khanate backed by the Ottomans, as serious regional rivals kn the 16th and 17th centuries. Though they had been subordinates to the Golden Horde, after the fall of the Mongol empires the Muscovites sought to represent themselves initially as first among the slavic duchies, and then as first among the remnants khanates as well, citing their patronage of the Chinggisid Kasim Khanate as proof of their right to an imperial, or khanate, title. As they eventually beat out the Crimeans for control of Kazan, Astrakhan and Bashkiria, also taking Sibir, they took the titles of Tsar and Khan simultaneously, though depending on whom they were addressing. In the 17th century pressure from the Kalmyk and then Kazakh armies led to more and more fortresses on these borders, preparing the grounds for even more expansion.

Finally, population pressure was something relevant throughout this period. As the Russian state deepened its institutions and agricultural productivity rose, peasants found themselves cramped and overtaxed, and thus searching for alternatives. The Siberian fur trade was one, but another were the fortified frontiers rich in trade, but also in plains customarily used as pastures by the Turco-Mongol tribes who lived on them, but possible to convert to farmland- which the peasants did. The authorities, eager to populate these frontiers to supply and maintain their fortifications, and not without an eye to expansion, incentivised such migration

Certainly there may be someone else who can give a more detailed answer, but this is what I know of. If you want sources for any of the points here I can dig through my older notes, just lmk.