When discussing Mongolia it's important to make a distinction between Outer and Inner Mongolia, the territory of modern day Mongolia was known as Outer Mongolia. Inner Mongolia on the other hand is currently a region of China. Inner Mongolia is much more densely populated and richer than outer Mongolia which is partially why inner Mongolia is still a part of china while outer Mongolia isn't. Since densely populated and urbanised regions are easier and more profitable to control since it's more difficult for insurgents to hide in those areas, whereas in sparsely populated area insurgents have plenty of space to hide. Also sedentary populations are easier to tax than nomadic populations. There was more incentive to rule over inner Mongolia than outer Mongolia.

The Qing dynasty which was a dynasty based in Manchuria (modern day north-east China) conquered Inner Mongolia in 1635, and later began their conquest of China in 1644. After conquering China they slowly started expanding their area of control into Outer Mongolia, eventually gaining loose control over most of it in 1691, although the region retained a lot of autonomy because it wasn't profitable to exert more direct control over the area, therefore outer Mongolia remained culturally distinct from China whereas inner Mongolia didn't.

In 1911 there was a lot of unrest in China and a revolution erupted with the aim of overthrowing the Qing dynasty, this revolution led to an erosion of the authority of the central government in China, which eventually led to the warlord era. After the Qing dynasty was overthrown Tibet and Mongolia declared independence from China.

In Mongolia Bogd Gegen, who was the 8th Jebtsundamba Khutuktu, the highest ranking Buddhist monk in Mongolia declared that the Qing used to rule over both China and Mongolia, but since the Qing dynasty was overthrown China had no claim to Mongolia. Bogd Gegen crowned himself as the Khagan (King) of Mongolia.

He managed to gain control over Outer Mongolia, but the Chinese government gained control over Inner Mongolia after two years of fighting.

In 1919 a Chinese force invaded Outer Mongolia and occupied the densely populated parts of it, however they were eventually driven out in 1921 by Roman-von Ungern Sternberg, a Russian general who fought in the Russian civil war. He was eccentric, Buddhist, and fascinated by Mongolia.

In 1921 the Mongolian Revolution happened, the revolutionaries defeated and killed Sternberg and turned Mongolia communist. In 1924 Mongolia adopted a Soviet style constitution and became a puppet of (but not a part of) the Soviet Union, with the collapse of the Soviet Union and the Mongolian revolution of 1990, Mongolia became completely independent.

In summary Mongolia is independent because it was poor, a vassal of the Soviet Union, and because it's considered internationally unacceptable to invade countries nowadays.