This has been bounced my way from a couple of sources lately:
To be clear what Alyssa is doing isn't literally saying the words backwards; with a couple of exceptions if you reverse the audio you won't hear the words as they normally sound. What she seems to be doing is pronouncing the words as if they were spelt backwards (there is a good rundown from my layman's perspective on the various backwards ways here).
This viral video I've found more interesting than most, because I can do it too. Not as conversational-speed fast as Alyssa can (and even if some of the words or combos are ones she has already figured out and thus knows from memory, the response speed is seriously impressive), but usually within the 'three seconds' yardstick for the video.
It is something I used to do as a party trick in my teens and twenties, but had forgotten about until now. If I can spell a word forwards, I can say it backwards. I just visualise the written word in my head and read it backwards. I don't jumble the letters until they read forward just reversed - reading it backwards is a lot simpler. In doing this I find I treat the word as if it is one I have never read before, breaking it into syllable sized pronunciation chunks, before saying it out loud. Like any reading, if you already know what a particular letter grouping sounds like you can just plug it in and make the process easier (some words also have natural backwards analogues, like time / emit).
I don't remember learning how to do this; I think I just tried it one day and found I could, and I've got no idea what this says about the way my mind works. I'm told though by people who know that the above is a reasonably tricksy cognitive ability, even if it seems perfectly natural to me.
After doing a little reading around on the topic I have been playing around with the other expressions of this, dna dnuof I evah on elbuort ginpyt sdrawkcab rehtie (or more correctly rehtie sdrawkcab gnipyt elbuort on evah I dnuof dna) albeit at a slightly slower pace as typing normally. I tried reciting the alphabet backwards this morning too, and found I could do that too. Reading mirror or upside down text is also not a problem. So can my wife, which is handy since she is a teacher. She can also do the reverse pronunciation a bit as well.
While I figured from the responses I got years ago that not everyone could do this, I didn't think it was that unusual that I could, and I know there are others that can do it a lot better. As it is I'm a bit bemused to find something I can do going viral. It's pretty cool. That may have been my fifteen minutes of fame right there :).