They key qualifier is living species, many living species have relatively recent origins, that is exactly what you would expect in an ever changing ever evolving biosphere. That is like stating that most cars on the road are less than ten years old, OK, but that's not really a surprising statement. If you plotted a bell curve of how old all current animal species are the bulk would be fairly recent geologically speaking. Species have a lifespan. The average lifespan of a mammal species is around a million years so simple statistics will mean most living mammal species will be less than half that. No one is surprised when you say most of the data points in a randomly distributed set fall within one-two standard deviations.
In fact based on back of the envelope calculations, if the average lifespan of a species is ~ 1 million years, and we assume they are randomly distributed, (they are not, extinction events exist which would shorten it), then just based on the 68–95–99.7 rule, 95% of living (at or below the mean) species should be less than 320,000 years old.
Now the paper itself is meaningless, it's justification is largely circular: define species based on mitochondrial mutation rate, (which of course is a time based calculation), then use that definition to say most species fall within a certain time period. You can change the number of mutations in your definition of species to fit your conclusion. Even a species has existed a single unfractured population for millions of years will count as a recent species by that definition. It is circular reasoning, made worse by using mitochondria, which are not undergoing recombination so variance is basically only controlled by generation time. If a population splits into two or remains a single population its mitochondrial variance will remain largely unchanged. Interestingly they exclude the groups (plants, fungi, ect) in which mitochondria undergo recombination.
Really the only conclusion you can draw from the article is most journalists don't understand statistics or genetics.