The horse usually symbolizes the libido. (Sigmund Freud defined libido as sexual energy. Carl Jung defined it as psychic energy.) It brings unconscious psychic energies to the surface.
The horse as a symbol has many meanings, but I think that the link with death, the underworld and the feminine is the strongest.
Witches (feminine energy, anima), also called ‘Lamias’, ride horses. That’s why we have night-mares. These Lamias even change into horses. Also the devil and Hel, the Norse goddess of the underworld, ride a three-legged-horse. Hecate, the Greek goddess of the underworld, is sometimes depicted with a horse head and the Valkyries bring the dead on their horses to the afterlife.
The horseshoe casts away the evil spirits. That is why farmers used to hang those above their door. They would also put a horse skull on top of their roof so it could carry away the Lamias (the night-mares).
‘Mare’ is linked to the word ‘mort’ which means dead. ‘Mara’ is Slavic for witch and ‘mora’ is Polish for nightmare. ‘Mare’ is Latin for sea and ‘madre/mater’ means mother. This is of course also linked to the name ‘Maria’ and the word mama.