This is, obviously, a very big question, and could be broken into parts or perhaps asked in different ways. I’ll answer it in the most straightforward way: The purpose of art is to gift the present into the future in a manner that only art can, which is to breathe art's unique truth telling out into the world, so that nothing can be hidden, not now, not ever. This is why art is the most effective, important and valued means for humans to connect across the ages, but also in the present.
Today, right now, art gives its recipients a deep view into their very way of being. What could be more valuable than this ability for us to see ourselves, affirm ourselves, affirm our views, the ones that define us and that we ourselves, all of us in our culture, in every action and choice shape and mold into being? Art both shows us what those large cultural choices are and it is the very shaping of those choices. This is why art is always deeply political. Each thing made is a choice and therefore a voice, a way of shaping the present. This kind of revealing that is so unique to art cannot simply be a matter of feelings that are there for our enjoyment or pleasure; it requires us to think and to feel in much more important ways than mere pleasure.
Art requires us to work for it, to be alert, to think, as opposed to entertainment, which does not, unless we distinctly decide to academicize it. We have to ask (ourselves) hard questions with art. Art holds this unique and, in my view, most authentic “voice” for humanity. It tells us who we are both in the present but also for generations to come. It unifies us in this way. I don’t think it shows us much in terms of progression, because I’m not yet convinced humanity has progressed, but it certainly shows us who we are, who we were, who we may have hoped we are or were, as the case may be. Art shows us these things through a kind of materiality that escapes everyday grammar and language.
Art wholly evades language and lives in the realm of the felt, the done, the very visceral reality of our bodies. Art is therefore akin to birth and death and sex and bodily functions, specifically bodily-emotions; the art experience is the one place where mind and body unite in both pure emotion and pure “thingliness”, the singular purpose of which, in my view, again, is to affirm the human experience across time in all its struggles, permutations, conflicts, inner and outer turmoil and many conflicting ideas and actions both at the interpersonal level and all the way up to the macro-historical and societal level.
When I walk into a museum, no matter the exhibition, no matter the era, no matter the work, I see gifts for the soul, time capsules to be opened by those willing to do so. This is why the purpose of art, in my view, cannot be separated from the "act of art". Art is participatory. If the viewers are not changing themselves, thinking, feeling, opening their minds and receiving then art will serve them no purpose whatsoever. This is one of the great crimes of contemporary art, that so many of its purveyors would like to stand in between this experience of the viewer and the art object and rather than fulfill the respectable role of bringing viewers closer to the experience of art they seek to distance them, to make art an erudite and academic process, when it is absolutely not; worse yet are the artists who rely on these devices of which we sadly find many. That line of thought is, perhaps, for another set of questions...