I asked this question similar to this (only about Russia not China) and this is the response I got:

This is a popular question on this subreddit so feel free to do a search. The answer is that Russia expanded quickly into the relatively sparsely populated areas on northern Europe and Siberia that weren't claimed by any other Great Power states. The expansion was quite similar to Western European colonial expansion in the Americas, except the Russians didn't have to cross an ocean to do it. The primary motivation were economic/commercial benefits of the pelt trade and (later) land routes to East Asia. The Russians faced some resistance from the local peoples but the Russians were equipped with better gunpowder weapons, among other things. After the collapse of the Golden Horde (itself a remnant of the Mongol Empire) in the 15th century, there was no power in northern Asia that could match the Russians or wanted to claim that land, so Russians simply expanded without too much difficulty.

https://www.reddit.com/r/AskHistorians/comments/3u8ris/how_did_russia_get_so_big/

I'm not as familiar with China, but I can explain Russia. For one, Siberia is sparsely populated. Second of all, Russia is routinely on the brink of collapse throughout its history. Siberia is a key factor in this instability and also the reason Russia was never conquered.

Before Russia existed, the grand principality of Moscow had to pay tribute to the golden horde, an offshoot of the mongol empire. While the east was sparsely populated, the people who lived there were very strong militarily and frequently raided their neighbors. Think of these people like the Sioux or the Apache, except much stronger. War inevitably broke out and Russia eventually defeated the various tribes.

The way Russia handled the aftermath of these wars cemented Russian authority. The steppe people had been nearly been wiped out by warfare and disease, but that want enough to ensure submission to Russia. First, the locals were russified where possible by converting them to Christianity and establishing Russian add the sole administrative language. A series of small fortified cities were built in Siberia for a dual purpose; they helped keep the local areas under control and they served as a strategic depth for European wars. However, these cities were not self sufficient and require subsidies to be liveable. These subsidies continue to this day.

The importance of strategic depth in Siberia cannot be overemphasized. The Russian core lies in the eastern european plain and was susceptible to invasion by many powerful nations. Sweden, Prussia, Poland, Lithuania, Hungary, and Crimea were all major powers who had no natural barriers to the conquest of Russia. Russia's major strength was it's population and it's size. The Siberian cities were crucial to this. But these cities were a major drain on resources and could cause economic problems.

These economic problems coupled with large minority populations can spell disaster in a time of crisis. Russification helped with this, and the imperial russian autocratic style of government helped to suppress nationalist tendencies. But this could not last, and a combination of economic problems, nationalist movements, and war led to the Russian revolution.

It should be noted that during this revolution russia lost a considerable amount of land in the steppe. When the Russian government is weak it tends to lose land on its periphery, and when it's strong it reincorporates it. This can be seen seen in the aftermath of the Russian revolution and the collapse of the Soviet union. Places like Kazakhstan and Turkmenistan were fully independent for awhile, but quickly fell back into Russia's sphere of influence once russian leadership was strengthened. Russia's expansion was never as thorough as the US, so it has a much more difficult time holding onto the steppe.

The growth of China into a large country has mainly been due to its history - a cycle of a single kingdom that implements feudalism (to varied extents), which then descends in separate kingdoms, and then the strongest kingdom would seek to unite all the separate parts together again, increasing in size each time.

At the establishment of the Zhou Dynasty 周朝, king Ji Fa 姬发 rewarded the people who had aided him in overthrowing the Shang dynasty 商朝 with titles and lands to use as their own fief. These lands were each developed by their respective lords, and grew strong enough to sustain themselves with time.

As the rule of the Zhou kingdom was passed down from generation to generation, the control over the individual lords grew weaker. This was partly due to some of the Zhou kings' incompetence, but also because the widespread area these lands covered meant that the central government could not consistently monitor what was done. The last of the Western Zhou kings, King You 周幽王, had further alienated his subjects by making a fool out of them, so that when he was attacked, they chose not to come to his aid. Thus, the Zhou government loss control over the feudal lords completely.

Although Eastern Zhou 东周 was soon established with a new Zhou king, the actual power now lay in the hands of the lords. This spelled the beginning of the spring autumn period 春秋时期, when the feudal lords would war against each other for control over the lands, with the bigger states taking out the smaller ones. The entire period, as well as the following warring states period 战国时期, was filled with inward strife, but also outward expansion, as the states each sought to outdo the competition by gathering more resources and manpower.

When Qin 秦 finally managed to conquer all the other states to reunite the lands back under a single empire, the total land mass that the Chinese owned had grown much larger, mainly due to the expansion done by the smaller states during the warring periods. This pattern would repeat many times throughout China's history, especially whenever there was a change in dynasty (e.g Jin after the three kingdoms period, Sui-Tang after the Southern and Northern dynasties).

Another cause of China's expansion would also be invasions, especially from nomadic tribes. A key example would be the Xiongnu 匈奴 during the Han dynasty. Due to the multiple incursions upon Han territory by the tribes, Liu Che 刘彻 decided that the only way to stop their invasions for good would be to attack them at the heart of their own territory. Thus the Han territory spread further towards the desert during this period of time, as they sought to conquer the lands owned by the Xiongnu. However, the land gained through this method was more likely to be lost again whenever there was even a slight political instability in the government.

Short answer for China. The "China Proper", the part South of the Great Wall and East of the Tibetan Plateau, the area that is mostly associated with anything "traditional Chinese culture" was the result of thousands years of mixture of conquest and assimilation. The rest of the territory fell in and out of Chinese Control, but what cemented the control was the Qing Dynasty, the one you might of seen in "the Last Emperor". The ruler of Qing, were the Manchus, from North East of China Proper. They built militarism into their culture, and conquered the lands that was greater than that of the Modern China. When the last emperor abdicated, the succession state gained the right to all that land.

Amateur answering for china here. It is easy to explain why China is big now but somewhat difficult to pin-point how China got so big as China's borders changed through different dynasties. There are various times when the imperial court may claim certain territories to be theirs but had almost no real control over those regions.

Short answer: China is big right now, largely due to the rise of the Qing Dynasty. The "Manchus" located in current day northern china around the Heilongjiang invade China at the end of Ming Dynasty when the Ming government was dealing with major budget crisis, droughts, famine and a civil war lead by a farmer based rebellion movement. The Manchus of the time had a similar military set up as the mongols. They were mostly calvary based and almost all men were active soldiers. They quickly took over the Ming dynasty amidst Ming's civil war. However, it is notable that the emperor that was responsible for the expansion of the Qing dynasty's border to include current day Mongolia, Tibet and Taiwan is KangXi Emperor of Qing. During his reign, China's influence and military prowess in Asia was probably at its peak of recent history.