Falling death rates produce a population explosion, but only for a generation or two. After that, birth rates fall in line and things level off.
So the first part says population increases due to reduced death rate. The second clause that this effect is limited to about 30-40 years. The next sentence that birth rates "fall in line" which I read to mean decline to account for the reduced death rate. Finally "things level off" meaning that the rate of change of population is reduced to 0, a plateau on a population graph.
Which overall I read to mean that whilst there is a short ("generation or two") increase this is rapidly accounted for by falling birthrates and the overall population growth is near zero, ie population is not increasing.
By "level off" I meant "return to a growth rate comparable to before the boom", though in actually it sometimes ends up negative -- Japan and much of Europe is breeding below replacement rates (read: committing cultural suicide). But more importantly, I never said that this has played out everywhere in the world; in fact I've strongly implied the opposite.
Now about this "strongly implied the opposite" the area encompassed included at least China and India as subregions and didn't appear to have a disclaimer - if any region was implied then it would have to be Asia, but I assumed the whole world. Which words "implied the opposite"?
I think in Europe the situation is that endemic populations are breeding below replacement rates (good IMO) but that immigrant populations are cancelling out any benefit and themselves breeding way ahead of the endemic rate. At the same time the countries they are emigrating from are exploding in population and the emigration is not sufficient to level off the population - ie that Worldwide population growth rates are positive but as you note but that the native populations of certain regions are not contributing to the procession towards untenable over-population.
