The phrasing of that section isn't entirely clear, but I think the idea is a little simpler than what you're imagining.

What the article means is that if you have a single, very short emission of waves with a combination of multiple frequencies, then in principle an observer ought to observe all of the emitted waves arrive at the same time, regardless of frequency. (Traveling the same distance at the same velocity should mean arrival at the same time, after all.)

In this case, though, they see the radio-frequency equivalent of a short blue flash first and later a short green flash and then a short red flash (and presumably all gradations in between, all blurred into one event of gradually decreasing frequency). The notion (as I understand it: not my specialty) is that the longer wavelengths get slowed down a bit more as they pass through galactic and/or intergalactic dust, so their speed is lower.

Now, what you're probably wondering is, "Why do they think it's a single short event that's gotten 'smeared out' during travel rather than a longer event that continuously changed wavelengths as it happened at the source?" I don't know the answer for sure, but astronomers have a lot of practice in recognizing these things. One possibility is pulse shape: if the pattern of brightening and dimming of the "blue" light looks exactly the same as the patterns for the "green" and "red" light, it's more likely that it came from a single source event rather than a continuous process.

Steuard is right, and astronomers have good reasons to expect this behaviour - wave dispersion. The best link I can find from a quick google is this: http://en.wikipedia.org/wiki/Dispersion_(water_waves) -which refers specifically to dispersion of water (typically, ocean) waves, but the concepts are the same. (Edited to add) We expect dispersion effects in the case of fast radio bursts due to effects of propagation through the interstellar medium. Not my area of expertise, but see http://arxiv.org/abs/1310.8316 if you want a properly referenced starting point.

Maybe it's worth mentioning that we're all at least a little bit familiar with dispersion in other contexts: it's the same phenomenon that makes a prism work to split white light into colors, and that is responsible for rainbows. (The angle that light bends when passing through a surface depends on the precise speed that light of that frequency travels on each side. When different frequencies have different speeds, that's dispersion, and what started as combined white light splits into a different color at each angle.)

They don't get 'slowed down' more, they get deflected more. So, the redder light that does reach us had to travel further to get here.

That was one point where I wasn't sure quite how to talk about this. In the back of my mind, I have a sense that "gets deflected more" (meaning "longer distance") is actually referring to the same fundamental process that "gets slowed down more" (in the sense of index of refraction: v=c/n) refers to: either way, the time to pass through the medium is longer than it would take to pass through an equally large vacuum region. But I wasn't quite confident enough to phrase it that way in my main reply.

That makes sense. Mucho appreciado :-)

Dead on, good job.

Thanks! (I'm a physics professor, so explaining stuff like this is pretty much my job. But my specialty is string theory, which is pretty far removed from radio astronomy!)

The text you quote says is that if you turn your radio transmitter on, and broadcast on a lot of different frequencies, and turn your radio off - all the different frequencies arrive simultaneously too(since you transmitted them simultaneously). It will of course arrive some time after you transmit it, since the signal travels at the speed of light.

tldr; It's just saying that a low frequency signal travels just as fast as a high frequency signal.

Sure - but if you bounce a radio signal off a wall to the side, then turn it off and on to send a signal straight at an observer, the time difference to the observer is not the same as the one to the transmitter.