This is where your imagination comes in. Begin cutting away from the sides of the paper. Usually, cutting small triangles from the sides is easiest, but don't forget to try other shapes.
Be careful not to cut all the way from one side to the other, or else you'll chop your snowflake in half!
In this particular snowflake, I cut the top edge (that original angle that we chopped off) to make the points turn out differently, I added some spiky cuts around the middle there, and I snipped off the very bottom at an angle (which will make a star shape in the middle of the finished snowflake).
I don't use patterns for my snowflakes: I just cut as I go. My snowflakes are never the same because I don't usually plan out my snowflakes. Sometimes I find a new cut or shape that I like, and I may use the same technique on a different snowflake, but I don't copy the whole snowflake.