Lorimer B. Bailes M. (2007)
A bright millisecond radio burst of extragalactic origin
Full text:
http://arxiv.org/abs/0709.4301
The universe is full of stuff, although there are big gaps between the bits.
Some of the bits are electrons - and there are electrons between the stars and between galaxies just flying around on their own. The paper talks about this as the galactic and intergalactic plasma (p2 etc).
This is what they mean by the free electron content of the Universe.
Radiation interacts with these electrons (and other charged particles) ... the electrons are very sparse, but over very long distances the effects add up.
Basically it's like looking at a light through a mist - you can tell how away it is from it's brightness and diffuseness if you have some idea about how light is affected by mist and how dense mists are. The researchers are basically saying they don't know how far away the source of the burst was, but the character of the burst was consistent with the best current guesses for how the intergalactic plasma affects radiation. Ergo: extragalactic origin - but still close enough to see it as well as we did.
Similarly, you can tell that a light you see outside the window was not in you backyard, because it looked like it was clearly affected by mist and the mist in your backyard is too diffuse for that.