He's not saying that the extinctions aren 't happening. He's saying that we don't know that they're comparable to the big extinctions of the past. It's worth remembering here that there is a huge problem with comparing modern world extinctions with extinctions in the geological past. First, it is vastly easier to see and catalog species today than in the past where fossilization is an extremely rare event. We can't know what didn't get fossilized.
But creatures which weren't easy to drive extinct, due to numbers, longevity of the species, or widespread habitat, would also be more likely to leave fossils. So we also have that the fossils of the geological past come from species which are more likely to not be threatened by extinction than the usual species today.
Third, species have different meanings in modern and geological terms. Today, we can classify species based on subtle distinctions like behavior, coloration, habitat, and most important, DNA which usually are impossible to determine from fossil records. Fossil species on the other hand, are determined by rather crude morphology traits which can be fossilized. A fossil species is a much bigger grouping than a modern species.
So when you combine all these aspects, you get that extinction of a fossil species is a much bigger deal just on its own than extinction of a modern species and may represent in some cases the extinction of dozens of modern species.
I think a better measure here is extinction at the genus level. Genuses are more likely to have fossil records and we can speak of the relative decline of the number of genuses in a proposed extinction event.
When you do that, I don't think there is a serious comparison at the present between human-caused extinction and geological extinction events. I suspect most genus-level extinctions would be in large terrestrial animals, amphibians, and any genus of organisms particularly susceptible to local habitat destruction. You don't have large scale declines in the number of all land and sea genus-level organisms (which can be fossilized) as are present during major extinction events of the past.