I mean, I understand that they were good cavalrymen and fierce, but how did they manage to conquer that much land?
For reference, this is a map of The Mongol Empire at the time of Genghis Khan's death in 1227 and this is a map of The Mongol Empire in 1294
Most historians agree that the Mongols number just under 1 million in population during their time of conquest. It is commonly accepted that Genghis Khan did not wake up one morning thinking he might conquer the world. His first goal was to unite the scattered Mongol tribes. After this success, he was elected Khan in 1206.
Genghis Khan introduced reforms which some consider innovative. He broke with steppe tradition when he ordered his men to take a city, but not plunder until the attack was complete. He also mandated that the treasure was to be divided evenly between everyone. He commanded respect from all who fought for him and divide his armies into a decimal system. He had units of 10, 100, 1,000 and 10,000.
In addition to the decimal divisions, Genghis Khan's army used superior tactics. A regular army of the time would have two lines of infantry, a group of cavalry and archers on both the left and right flank, and a group of reserve infantry in the extreme rear. The Mongols had a small line of infantry (we assume that these were people who had been take captive from opposing people groups) and a line of cavalry who rode in a circuit. Steppe ponies, which were the mounts of the Mongols, are only able to run maybe 1 or 2 miles before having to rest. The Mongols brought with them 5-6 mounts. As the Mongols rode in a circuit with approximately 60 arrows in their quiver, they could ride around, shoot a the enemy, dismount, pick up another horse, and ride the circuit again. They were able to stand approximately 1 mile away from the enemy, ride toward the enemy, and still have accuracy even at 1/2 a mile or so. This meant that the Mongols never had to fight in close quarters with the enemy.
Once the unit cohesion of the enemy was broken, the Mongols would begin to ride through the enemy lines and take out infantry, cavalry, and archers. The Mongols never had to get off of their horses to fight. The Mongols were also very skilled in the Carpathian shot, or shooting an arrow backward toward the enemy when riding away. It also made no difference to the Mongols if they were able to defeat the enemy when in retreat or in advance.
The Mongols were masters of the feigned retreat. The sedentary populations did not understand the Mongols. If you are a sedentary infantryman guarding a city, you only want to chase the enemy away so that they won't bother you any more. The Mongols knew that they needed to either absorb or kill the entire army and the leadership of a people group to ensure that the group would not seek revenge. This was something that Genghis Khan learned from his childhood. The Tatar tribe killed his father when he was a young man. Genghis Khan's first act as Khan was to destroy the Tatars in their entirety. The Merkid tribe kidnapped his wife Borte to avenge the kidnapping of his mother Ho'elun. In turn, Genghis Khan eventually destroyed most of the Merkid tribe and absorbed the remaining populace into the empire.
I would argue that this last point is the most important answer to your question. The Mongols either utterly destroyed the enemy or absorbed their women, children, and some of their army. They killed off the leadership of a population so that they would not have further problems from that people group.
My sources for this are Jack Weatherford's Genghis Khan and the Making of the Modern World, Christopher Beckwith's Empires of the Silk Road, Urgunge Onon's translation and commentary on The Secret History of the Mongols, and a lecture in my graduate class on the Mongols.