That is an interesting article. In short, yes, they are for real. Speed training, or dynamic effort training, is an important part of some powerlifting programs, such as WSB. I think the best way of explaining this is through an analogy I read in a WSB article.
Take 3 objects: a ping-pong ball, a baseball, and a heavy rock (>50 lbs). Then throw all of these objects. Which object can you exert the most force on? Remember, F=m*a. And the answer is the baseball. The ping-pong ball is too light, while the rock is too heavy. But with the baseball, you can exert the most force.
I would guess that the explanation as for how this applies to strength training is that the faster you can accelarate a given mass, the more force you can exert. And the more force you can exert, the more weight you can lift. This is just my guess, but I do know that it is important for strength training. I know Aaron_F does explosive push-ups as part of his training, so they must be good.
My only concern would be that push-ups are too light. WSB recommends doing speed bench with 50% of your max. I worry that explosive push-ups are like the ping-pong ball in the analogy. Chad Waterbury makes an interesting argument in their favor when he talks about how when doing dynamic bench presses, the lifter has to slow down the weight at the top or he will destroy his joints.
The 1 minute rest period is also found in WSB. Dave Tate says that it is crucial that rest lasts for 1 minute, no more and no less.
To wrap it all up, I would say that explosive push-ups are definitely good, but speed bench might be better.
