This is still an area of active research [1,2]. The basic hexagonal symmetry comes from the hexagonal lattice of bulk ice. What is more intriguing is the global symmetry between individual dendrites. From reading some recent papers, I think the mechanism is as follows. The snowflake is small enough that although the temperature and water pressure surrounding it can change quickly, all its dendrites are always exposed to almost identical conditions at a given time, leading to a largely deterministic growth, hence the symmetry.

[1] https://www.nature.com/articles/s41524-017-0015-1

[2] http://iopscience.iop.org/article/10.1088/0034-4885/68/4/R03