Generally the mechanism goes by the name of "spontaneous symmetry breaking". This is a very general mechanism, and occurs in many situations. The super short description is that at high temperatures, the symmetry is observed. But at lower temperatures, the bits of space-time near one another tend to prefer to have similar configurations, so that regions of space-time self-organize into the same configuration. The precise configuration they organize into is random, but it is the same across the local region.
This is analogous to a magnet. Terrestrial magnets at high temperatures don't produce any significant magnetic field because the atoms within them are oriented in random directions. But as you drop the temperature, those atoms like to line up so that their individual magnetic moments point in the same direction. So as the metal cools, those atoms line up together. At high temperatures, there was a symmetry in that no particular direction within the material was special, but at low temperatures there is a magnetic field which picks out a particular direction, breaking directional symmetry.
