The six-fold symmetry of a snow crystal ultimately derives from the hexagonal geometry of the ice crystal lattice. But the lattice has molecular dimensions, so it’s not trivial how this nano-scale symmetry is transferred to the structure of a large snow crystal.
The way it works is through faceting. No long-range forces are necessary to form facets; they appear simply because of how the molecules hook up locally in the lattice (see Crystal Faceting for how this works). From faceting we get hexagonal prisms, which are large structures with six-fold symmetry. Eventually arms sprout from the corners of a prism, and six corners means six arms.
Faceting is how the geometry of the water molecule is transferred to the geometry of a large snow crystal.