Why does ouzo turn milky white when water is added?

It's an emulsion. The oil which makes the delicate flavours of ouzo dissolves very nicely in alcohol, but it doesn't dissolve very well in water. When you add water, it binds to the alcohol, which dissolves well in water. The oil eventually stops binding to the alcohol and starts hanging around with other oil molecules. These little aggregations of oil in the water make the drink turn milky white. If you were to zoom in with a very powerful camera, you would see these tiny globules of oil being completely surrounded by water molecules. This is exactly how emulsion paint works, where you have globules of oily paint molecules surrounded by water.