By the wind. You know the little puff balls that form in the fall? well the seeds have small bits of fur on the top of them almost like wings so that they can float on the wind and be scattered, therefore spreading.
Mix your dandelion seeds with sand to make it easier to disperse them. To get the best results, use sterile sand at a ratio of about one part seeds to three parts sand. Put the mixture in a container and sprinkle them over the soil evenly using a slow, even movement. It is better to sprinkle fewer seeds at the start to ensure you have enough seeds to cover the entire plot.