The answer is that it's more of a psychological solution, rather than a mechanical one -- It helps the players think like a team rather than individuals, it helps them pay attention between turns, and it helps stay excited about the game. Because of that, the combats move faster -- the characters are more effective, monsters die faster, players have their actions ready when their turn arrives, no one is bored and needs to be spurred into action.
There's more to it than that though. It's not simply a fix DM's can use for bad/easily distracted players. I know you aren't saying exactly that, but your responses read like it. It is a method (in my experience) that benefits groups of any level of experience, and in more ways than simply psychological.
A few of those "other" ways...
1) This system does not track initiative order. The standard one does. It may not seem like much to have to make up cards, and re-arrange them regularly as needed, or manage a whiteboard list erasing and updating as needed, but there is a big difference between having to do that, and not (I have done many of both). Right now, my initiative tracking time takes me no time at all. I roll once... who beat it? Then you go now. That's it - it's all me / them / me / them after that, the order inside each of which does not matter, even with readied actions.
...This makes things much faster not only for me, but for the players
2) When I manage the monsters, it's MUCH faster to do them all at once, as opposed to one, then taking inventory of what a PC does, then a few more monsters, then tracking what another PC or two does, etc. It's an efficiency thing, and I think most people will tell you (and you probably already know) it's much more efficient to do one task at a time (managing all my monsters at once) than it is to be disrupted and have to break that one task into many pieces done between other tasks. We have software at my company (and it exists in many more) meant to prevent just this type of thing from occurring.
...This makes things MUCH faster for me
3) The players get to be much more strategic and work together on their planning. The Wizards Thunderwaves some goblins into a pile, the Sorcerer nukes the pile, and the Fighter jumps in using Sweeping Blow to mark them all. With the standard init system, The Wizard pushes them into a pile, and then the goblins may get to go... so they move in to attack... now the Sorcerer can't do the blast he wanted to... and the Fighter needs to think of something else because now that it's his turn, things have changed dramatically. But remember, the monsters have the same benefits of going all at once
...This allows the PC's to do some really cool combo-like things they otherwise do not get to do (most of the time).
If you (anyone) try it for a while, you will see it is not the same.
I'm not saying it will always be better... maybe you will like the standard method better, but the point is it is not the same thing. It's not just a fix for poor playing or distracted players either.
Whimsical asked...
Does anyone have any more information about this method? I don't want to try it until I'm ready to handle the peculiarities of this method. Anyone else use this method? Let me know about your experiences and feedback about it
I say give it a shot, it's awesome (imo). It has done a lot for mine, and many other games - it's sped them up, got players much more involved, allowed for much more dynamic strategy, makes my job much easier/faster, so on and so on
If you have specific questions about it, post em here - I will do my best to answer what I can based on my experience with it.