Dont they need some kind of negative or exotic energy to do that? And that actually exist? How do you create one?
Exactly.
This is just a version of an Alcubierre drive:
http://en.wikipedia.org/wiki/Alcubierre_drive
Now, the Gizmodo article was very misleading. There were A LOT of problems besides just the energy problem being discussed here. For example, there was also the exotic matter issue that you correctly bring up.
There are fairly strong theorems which indicate that in order to create a warp bubble, you need exotic matter. As defined here, exotic matter is matter which violates laws of physics (essentially negative mass in this case). Thats a bit of a problem.
Now you can do some handwaving and hope that quantum mechanics lets you work around this. After all, you can play some games with the Heisenberg Uncertainty Principle and get local areas with "negative" energy. But then you are combining quantum mechanics and general relativity, and we don't have a theory of physics which successfully does that.
Thats not to say that any of this is a waste of time, of course. Just that I wouldn't exactly get my hopes up. The chance of any kind of payoff is extremely low since you would need some kind of new physics for this to work.
To answer the question of how this experiment is trying to get around the exotic energy requirement:
There are a few approaches you can take. The most likely thing to do is to create negative pressure. For example, something like the Casimir effect might do the trick. I don't think that you could really extract such energy though...
Reading the author's paper, it looks like they are trying to use dark energy. Dark energy is essentially a negative pressure term in the expansion of the universe. They are trying to use this to create the warp bubble. Yeah...thats pretty speculative on their part.
(I don't think that dark energy would ever work for spaceship since its energy density is so low, but there are about a billion hurdles before that would be an issue)