This experiment can be a little tricky to understand, but it all involves a property of water called Surface Tension. Water molecules are attracted to one another but not in the air. The water molecules will try to make the surface as small as they possibly can to be close to one another; this is what surface tension is. Milk is in fact mostly water, so it, too, has surface tension.
The washing-up liquid is designed to break up the surface tension of water so it can dissolve and clean fats and grease in the kitchen. As a result, we can use washing up liquid's surface tension-breaking ability to break up the surface tension of the milk!