What 33 species have cats made extinct

The first claim in your article

The authors estimate they are responsible for the deaths of between 1.4 and 3.7 billion birds and 6.9-20.7 billion mammals annually

comes from this article (full-text), where the authors state (table 2) that domestic cats are responsible for the deaths of 1.4-3.7 billion bird deaths and a much larger number of mammal deaths in the US and Europe combined. They also state that the majority of these deaths are caused by unowned/feral cats.

This is a meta-study examining the existing literature on domestic cats with some novel analyses, and assuming that you agree with their analysis, this claim is true.

The second claim is that

Felines accompanying their human companions have gone on to prey on the local wildlife, and they have been blamed for the global extinction of 33 species.

is a slight misinterpretation of a statement in this (full-text) paper that states

Based on our database, feral cats on islands have contributed to 33 (13.9%) of the 238 global bird, mammal, and reptile extinctions (including species extinct in the wild but extant in captivity) recorded by the IUCN Red List.

What this says is that cats have contributed to, but are not solely responsible for these extinctions. Be aware though, that the paper is talking about extinctions in island fauna, which are often a special case because of restricted ranges and limited population sizes, especially when it comes to endemics. Such extinctions have been ascribed to several introduced predators (rodents, ferrets/minks), not just cats.

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1. *Number of studies found that include an estimate of the model parameter.
2. †No calculation was conducted for this data grouping because of limited data.