Symbolism is the recognition of the ontological correspondence between a given reality — usually natural or at least accessible (water, a tree, a bird, the number two, a cross, etc.) — and another reality that surpasses us.
This correspondence is founded on the bond that runs through all states of existence: corporeal, psychic, and spiritual.

In practice, symbols, by their anagogical power — whether contemplated or enacted (e.g., in rituals) — lead to higher states of understanding.

More precisely

Symbolism is the act by which the mind recognizes, in sensible realities, the signatures of the invisible world.
It is not a subjective projection but an objective reading of reality, since the correspondence between visible and invisible proceeds from the analogy of being: all things participate in a single Principle.

Symbolism thus follows from the very structure of reality, understood as a fabric of vertical and horizontal relations, each level reflecting the others analogically.
Hence, the world is not merely a collection of objects, but a book whose symbols constitute its letters and sentences, opening the intelligence to its spiritual dimension.

One may say that symbolism is at once:
cosmological (the world is symbolic);
psychological (consciousness is symbolically structured);
metaphysical (the symbol refers to its archetype).

Symbolism therefore goes beyond any aesthetic or merely cultural approach: it concerns first being, its structure, and the participation of creatures in their model.

Further reading

Plato, Timaeus; Phaedrus — cosmos and anagogical images.
Hermes Trismegistus, Corpus Hermeticum — universal correspondences.
Pseudo-Dionysius, Celestial Hierarchy — liturgical and theophanic symbolism.
Thomas Aquinas, Summa Theologiae — analogy of being.
Meister Eckhart, Sermons — symbolism and the birth of God in the soul.
Jacob Boehme, Mysterium Magnum — signatures of the world.
Goethe, Theory of Colours — phenomenon as symbol.
Jean Borella, Symbolism and Reality / Crisis of Religious Symbolism / Histoire et théorie du symbole — ontological foundations; anagogy.
Bruno Bérard, Metaphysics for Everyone — symbolism and participation.
Henry Corbin, The Paradox of Monotheism — imaginal and symbolic.
Mircea Eliade, Patterns in Comparative Religion — structures of the symbolic sacred.