Here's a video on how Korean bows are made (press CC for subtitles). The construction is almost identical except the Koreans used bamboo and water buffalo horn while the Mongols used birch and ox horn for the core base and the belly respectively

Essentially they're a wooden base with laminates of horn on the inside/belly and sinew on the outside/back. The reason this creates a more powerful bow is because horn is more flexible in compression and sinew more flexible in tension than wood.

While a wooden bow is almost at it's limit just being drawn from being straight to full draw length, composite bows have so much 'extra' flexibility left that they're usually designed with heavy reflex. That is, the limbs start forward of the grip and have to be pulled back quite a bit just to string it. This is a bit like having extra draw-length which means more energy.