Jenkins is build server software. It's existence is related to the story only because it was a Jenkins developer.
'git push --force' pushes your copy of the repository to the remote, regardless of what the remote has. He basically told GitHub, "Hey here's the /real/ copy of the repository, the copy you have is totally wrong" and it overwrote all the history because his copy was a few months out of date.
I don't really know what Jenkin is and what "git push --force" does.