Just a suggestion rather than a definitive interpretation of her meaning:
Real "universal" as meaning "for everyone," and in each sentence, read the second term as referring to what is possible to experience. That is, science aspires to provide an understanding of things that anyone could, in some sense, experience were they at the right places at the right times. Indeed, it tries to give an account of what everyone would experience if they were at the right places and times and had the right equipment, senses, memory, etc. Art, in contrast, draws on private experiences — the stuff of our private thoughts, feelings, dreams, and imaginings. If art tries to provide a universal experience of those things, that means that it's a matter of sharing private experiences with everyone.
Now, I am not sure why we shouldn't think of science as also aspiring to providing understanding for everyone, and I am not sure what to make of art that tries to capture ideal types of things as they're experienced by everyone. (That is, I am not so sure art necessarily starts with the personal.) But you haven't asked whether her claim is plausible!