In this excerpt, elephants are made to look at themselves in a huge mirror. Other animals would normally ignore their reflections or simply think of these as other animals. Elephants, on the other hand, seem to be self-aware. They seem to recognize the reflections on the mirror as themselves. That's why they act in ways that are similar to the way humans look at themselves in the mirror.
Another experiment aims to find answers to the question "Do elephants think about what other elephants think?" When the experiment was done, it does seem to suggest that they do.
Another very interesting aspect of the experiments is when they are used to find out if elephants have the mental capacity to be spiritual or sort of come up with something akin to religion. Elephants are social animals. As the clip points out, there are anecdotes concerning elephants visiting the places where relatives have died and possibly reflecting on them. These are of course still very difficult to understand and to come up with conclusions will be very premature. But that's what science is all about. To seek knowledge on how these things work.