A 'transformative experience' is, according to Rebecca Chan's recent SEP article on it, is an experience that transforms you epistemically and personally; that is to say, in the experience you learn 'what it is like' in at least way that you could not have known before, and your values and perspective shift in a major life-guiding way. Currently the idea is a hot topic in certain sectors of philosophy, largely because of L. A. Paul's arguments that rationality, in the sense used in decision theory, cannot apply to transformative experiences; this would be very significant, because it seems that transformative experiences are extremely common, and, indeed, it may well be the case that any experience is potentially transformative in this way. Part of this may be the inevitable looseness of the characterization -- on the epistemic side, every experience is generally held in these sectors of philosophy to have a 'what it is like', and there is a general problem with knowing 'what an experience is like' without the experience; on the personal side, any experience could in principle touch on some core features of one's view of the world. In any case, I actually want to use a much stricter account of 'transformative experience' here, although having the same structure. By 'transformative experience' I mean an experience that
A) transforms you epistemically in the sense that you understand things that you could not understand at all without it;
B) transforms you personally in the sense that you receive new life-guiding values and perspective that you could not have at all without it.
In this very strict sense of 'transformative experience', it is fairly easy to argue that the Platonist account of philosophy takes it to be a transformative experience.
We see this in the Allegory of the Cave, which depicts a philosophical moral education. We all start in the Cave, watching shadows on the wall. But then we stand up and look around and begin to make sense of those shadows in terms of something much more fundamental. This continues as we go out of the Cave toward even more fundamental things, until we hit the Sun, which is the Good, on which all else is based. In this picture, philosophy is portrayed as an epistemically transformative experience. Not only does the person who leaves the Cave learn new things, he discovers when he comes back into the Cave and tries to talk to the cavedwellers that they literally cannot understand what he is talking about. No matter what words he uses, they relate those words to shadows on the wall; when he describes his experience, he cannot use any words that they don't relate to shadows on the wall, and therefore he seems to them to be speaking near-gibberish about the shadows, or else to be using words in a way in which they take as not meaning anything at all, or else to be making up fictions about shadows that aren't the shadows that they all see. To stand up and see the puppets making the shadows, to leave the Cave, creates an epistemic divide that the cavedwellers cannot traverse unless they, too, stand up and look around and leave the Cave.
Moreover, the Allegory of the Cave depicts philosophy as a personally transformative experience. The purpose of the Allegory is not to talk about reality in general (although it has implications for reality in general), but about moral education in particular. To stand up and look around changes your values from focus on sensible goods (pleasures and pains and the like) to focus on something more fundamental (civil or social goods, perhaps); to leave the Cave involves a shift in values to goods that are more fundamental (virtues) and ultimately to the Good, at the edge of what we can actually know. To leave the Cave is to become a changed person. And Plato actually emphasizes this by suggesting that the cavedwellers are like the dead. They live in a moral underworld, and Socrates borrows a saying from Homer about the shades in the underworld (which he had previously criticized if used about the actual underworld), "Better to be the poor slave of a poor master than to live as they do." The one who has left the Cave can no longer play their shadow-games about value, on which depends the only standard of success that they prize, because they are mere shadows to him. The mismatch between their values, i.e., what appears good to them, and real good, is too great for him to take seriously or even think of things only in terms of the former.