How does art mean?
This question will be answered differently for the different arts, though the different answers may have some things in common. Literature, for example, carries at least all the sorts of meaning that language carries. When this includes metaphors and symbols, or when rhythms, rhymes and sound play become important parts of it, then it may share modes of meaning with music, painting or film.
The nature of literal linguistic meaning is itself no easy thing to pin down. Philosophers and linguistic theorists are not agreed on it. Metaphor and symbol are even more difficult. (Link to theories of Richardson, Max Black, Ricoeur, and others).
But what about the other arts? How do they mean? Or do they do so at all? The composer Stravinsky is famous for claiming that music does not mean anything, it just is. Any meanings are brought to the work by the listener. (Link to this.) By contrast, jazz musicians put a high premium on "saying something" when you play.
Theories of meaning in music, in painting, in architecture, ...
Langer, Stravinsky, 3 essays on musical profundity, Kandinsky and the Blue Rider Group on the meaning of color, Jungian theories of archetypal meaning, Freudian & Lacanian theories, basic visual/musical/physical vocabulary, etc.