As a native English speaker, if I were to encounter that phrase without any other context, my first reaction would be to think that whatever video being discussed was very emotionally moving and/or intellectually pertinent. Much in the same way, some speeches are said to be powerful (one could say "That was a very powerful message" for instance).
Neither are necessarily powerful in any tangible way, but they are powerful in the sense that they can change people's moods and opinions of things. Change how someone sees the world, at least temporarily.
For an example of a video (it also happens to be a speech, but that's beside the point, the visuals and acting are truly what makes it powerful instead of preachy) that I'd easily be able to characterize as powerful watch this clip from The Great Dictator. Another video I'd also be inclined to call powerful would be the docking scene from Interstellar.
Of course, it's important to remember that all this is subjective and someone might well find a simple video of a sunset to be equally powerful. I just have a penchant for the grandiose.