I can see a country so vast in size grow like that, but I was under the impression that Russia was occasionally going through bad harvests and famines.

In 1896-1897, the number of births per 1 woman in Russia was 7.06. In Russia there were 50 born per 1000 population. Traditional Russia was characterized by unusual for Europe and high even by the standards of agrarian civilizations birth rates and the number of births per woman. This was facilitated by the early age of marriage, the lack of tradition of birth control and high marriage rate. At the same time, mortality was also high: in the 1870s, about 37 deaths per 1000 people. From the end of the XIX century, mortality in Russia begins to decline: 1890 - 36.7, 1900 - 31.1. After the revolution and the Civil War, the decline in mortality continues: 1926 - 19.9. Moreover, the number of births per 1000 people is 43.3 in 1900 and 43.6 in 1926.

An unusually high birth rate by European standards, coupled with a gradual decrease in mortality associated with improved sanitation and the quality of medical care, starting in the mid-18th century, leads to abnormally high growth rates of the Russian population in relation to the rest of Europe. Between 1750 and 1850, the population of the Russian Empire grew 4 times: from 17-18 million to 68 million people. This increase in population was largely due to territorial expansion. However, even with the exclusion of this factor, population growth rates are significantly higher than those that were then characteristic of Western European countries. Since 1850, when the territorial expansion of the empire more or less ended, rapid population growth has continued (68 million in 1850, 124 million in 1897).

The growth rate of the Russian population at the end of the 19th century - 1.2% per year - was high by any standards. Only countries that attracted emigrants on a massive scale, such as the United States, had a more dynamic population growth.

From the late 1920s - early 1930s, the situation has changed radically. Socialist industrialization assumed an unusually early, in terms of the level of economic development, involvement of women in employment outside the household. The involvement of women in employment entails a parallel process - a decrease in the number of births per 1 woman in the second half of the last century.

What makes even less sense to me is that the population grew very little since then.

The population of the former Soviet Union is 298 million. You can add the population of Finland and Poland to these numbers if you want.

Source: Moon D. The Russian Peasantry 1600-1930: The World the Peasant Made., 1999.

This is a little off topic, but I'd just like to address that the Red Army didn't win WW2 because of numbers alone. At the start of Operation Barbarossa, the number of Axis forces in the East outnumbered the Red Army, as did the total population of the countries that fought against the Soviets (though population isn't always the best method of measuring manpower availability). The Soviets didn't gain a manpower advantage until (approximately) 1943, after they had already begun to push the German Army back.

There's also another myth, similar to this, that they won because they could afford to take massive casualties compared to the Wehrmacht, but that's simply untrue. In terms of deaths in combat, they only lost a little under 7 million men compared to the Axis' 4.5. If you discount Operation Barbarossa, when the Red Army was massively unprepared for war, the death rate for the rest of the war is only ~6 million to 4.5 million, a ratio of 1.6 to 1. Soviet casualty rates are often over-reported to ~10 million, because 3 million Soviets died in German captivity compared to only ~600,000 Germans who died in Soviet captivity. The reason for that has nothing to do with combat performance - in fact, the Soviets actually took more prisoners than the Axis did - but because the Germans horribly mistreated prisoners that they took on the Eastern Front.