How are biomes formed?
The same way species are formed: evolution
Biomes are essentially a collection of all the ecosystems in the world that exist in the same climates and environments. All the desert ecosystems in the world make up the desert biome, for instance.
Each climate creates a selective pressure for any organism living in that climate. Arid climates select for plants and animals that can both operate on little water, and retain that water well. Thus, no matter where you are in the world, every desert is going to exert the same selective pressures on organisms. Eventually, natural selection in each independent climate and region creates similar looking ecosystems and trophic levels due to convergent evolution, leaving you with a biome.
Biomes are formed around similar climates. Since similar climates will all exert the same selective pressures, the ecosystems in each climate will all resemble each other somewhat. These can be grouped into biomes.