MIP levels are different textures in the same texture file, that you render according to distance from camera. It allows you to show one texture far away, and another more detailed up close. In the video, Frogboy replaced the "far away" texture with some random stuff.
They are commonly used in Source games as a way to do funny sprays that show one image far away, and another up close. For any other purpose than saving memory, their use is limited, because people running at low graphic settings only ever see the low-quality far-away texture.