As with most remote exploits targeting a 3rd party plugin (i.e, Adobe Flash, in this case), the attacker used a malformed compressed flash video to corrupt the embedded flash player on the victim.
The exact mechanisms are unclear to me as I have no intent on clicking on the video you linked because that video COULD be malware itself... however, the mechanisms are always the same: at some point a data buffer of some size that was improperly parsed was able to override some key values in memory that resulted in either data leakage or remote execution on the victim.
The fact that it was a Flash movie is not surprising in and of itself. Some older vulnerabilities in Flash player, that I'm aware of, involve dereferencing invalid blocks for the size of the images embedded in the video and drawing outside the allocated canvas region. Basically, the attacker changed the dimensions of the image/video such that they were valid according to the flash scripting language but were invalid in practice and thus when the flash player tried to assemble the content it corrupted memory.
What's particularly fun/interesting is the targeting of Linux - because Windows is the dominant target for these kinds of attacks. Linux tends to be more challenging because users are typically more cognizant of their threat level and take precautions. That's obviously not a golden rule by any means, but if you're playing around with Linux you are at least trying to learn how systems work much more so than being a straight Windows user.