Everything in the universe is spinning, including galaxies. Galaxies start out as big spheres of gas. All those particles are orbiting around the center of the sphere. Some of them might be orbiting almost vertically, while others are orbiting more horizontally. Over a very, very long period of time, these particles crash into each other, and when they do, they cancel out the different directions that they were going in and start going in the same direction (conservation of angular momentum). So eventually, most of the things in the galaxy end up going in the same direction, because the stuff going in different directions crashes into each other.
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