It's hard to deny the overall concept that our influence on the planet is causing a similar extinction event similar in size to these. Kolbert focuses on a number of particular research groups studying a particular species or groups of species that is under threat. These are really great looks at how science operates, and what scientists do, and paints a vivid picture of the facts and conclusions. At the same time, I think that in addition to these embedded journalist pieces, the book could have used a better summary and conclusion to tie everything together to suggest the overall impact. But perhaps that's just too huge to tackle and loses some immediacy in its hugeness. People will care more about that adorable squeaking bat over there suffering from white nose syndrome than about "37% of all species on earth".
There's also good variety among the cases and causes. Ocean acidification affecting corals, predation of moas and other prehistoric fauna that isn't around any more, climate change shifting habitable zones -- if you live on flat land, and the climate warms up, you can move north to stay at the right temperature. But if you live on a mountainside, all you can do is go up in altitude. Not only does the temperate zone shrinks, but if your mountain isn't tall enough, it vanishes.
Now that humans have air travel and poke our noses all over the place, things come along for the ride. Species that never would have run into each other do so now. Sure, we know about how smallpox affected Native Americans, but this story is being played out many times over, including the white nose syndrome decimating some bat populations in North America, caused by a European fungus that doesn't much bother European bats.