It's like "No no no, she has no cats, what on earth you're talking about?" Or: "No, of course she does not have cats, rest assured". Just an emphasis :-)
Why can't you say she doesn't have a cat? I know that кошек is genitive plural, but "she doesn't have a cat" and "she doesn't have cats" mean essentially the same thing (unless this sentence means she could have one cat, but not many "cats" but I took it to mean she doesn't have a/any cats). I think both should be accepted when translating to English.
I would say learning the gen. plural of "cat" is the main idea behind this sentence. Besides maybe in Russian this sentence makes total sense. I know at least translating it literaly to Spanish does.
Interestingly, in russian "у неё нет кота" has the meaning of "she does not have THE cat", at least to some extent