Basically the entire southeastern half of modern day Ukraine, which is among the best cultivatable land in the world, was uninhabitable because it was naturally steppe and living there put you at constant risk of raids from horse nomads (Crimean Tartars).
And because it was uninhabited, regular armies could not cross it (due to lack of forage), so the Tartars could not be atta Ked on a large scale.
This isn't really true. They also had a fairly advanced siege corps, utilizing cannons and bombs to take down fortifications.
This played a huge role in their ability to conquer. The domination of horse archers was nothing new but traditionally you could hide behind walls and starve them out. Feeding ten thousand besieging troops is hard enough without thirty thousand horses.
With the Mongols able to destroy walls, armies needed to fight the pitched battles that horse archers excel at.
The Mongols could literally run circles around other armies, avoid battles when the circumstances were unfavorable, and strike weak points deep in enemy territory.
Even if other armies had horses and bows of the same quality, they could not easily replicate this because their structure, in many cases even the social structure of the societies that fielded them, was based on cavalry being heavily armoured elite shock troops rather than archers.
Conversely, the Mongols fared poorly in areas where there was insufficient grazing for their horses, and could be defeated by appropriate tactics when they did not have the option to choose their battles as easily.