The Science of Milk kaleidoscope
Milk contains a lot of things, but two of its main components are water and fat. The two are mixed together well in what is called an emulsion (which is where the two liquids are mixed together, but it you could could look closely enough, you would find that there are lots of tiny little bits of each).
The fat gives the milk a high surface tension, so the food colouring floats easily on the top and doesn't spread very much.
However, the washing-up liquid tends to separate the fat and the water a bit (in other words, it "de-emulsifies" the milk a bit). That means that the surface tension in some parts is weaker than others and so everything starts to move around in swirling patterns.
The food colouring follows these "swirls" giving the kaleidoscope patterns you can see in the video.
If you want to try this yourself, you need very high fat milk (Fletcher uses high fat milk with a bit of cream added to it). Also, don't drink the milk afterwards - it will taste horrible!