C.S. Lewis is pretty well-known for his ‘argument from desire’, which is more or less a take on certain aspects of Platonic philosophy. Nostalgia and joy (or sehnsucht, or longing) for Lewis are indicators of our other-worldiness. Our desires and longing for beauty reflect our desire for the Divine, beauty itself. That small nostalgic ache we get at the end of a beautiful symphony or as a sunset fades is a desire for something ‘which no natural happiness will satisfy.’ Sensible beauty serves to awaken a much deeper longing for the beauty of the absolute.
Augustine makes a similar point in with the famous opening phrase of his ‘Confessions’: our hearts are restless until they find their rest in you. Pascal says similar things as well. Beauty, the experience of the beautiful, our desire for the beautiful, is a reflection of our desire for the absolute, for beauty itself. Lewis makes another point that when we have summoned into glory, that old ache will be healed. We’ll be made whole again. (A quick Google search will yield a large number of quotes made by Lewis on this topic that are worth reading)
‘The beautiful is unquestionably a transcendental orientation of the mind and the will, because the desire it evokes can never be exhausted by any finite object; it is an ultimate value that allows one to make judgements of relative value, and that weds consciousness to the whole of being as boundlessly desirable. whether or not there is actualy such a thing as an eternal beauty beyond the realm of the senses, the effect within us of beauty’s transcendce as an ideal horizon, toward which the mind is habitually drawn and apart from which the mind would not be open to the world in the way that it is. And that, in itself, is enough to render the physicalist narrative of causality profoundly dubious.’ (David Bentley hart, ‘The Experience of God: Being, Consciousness, Bliss’, p. 285)
Now, obviously, this isn’t an argument of any kind – nor is it ignoring the physiological/biological aspects of nostalgia, which, incidentally, is a fascinating study. Think of this as simply some musings on the transcendent nature of the experience of beauty.