The stag, in various forms, is an important symbol in many of the world's religions and traditions. 
In many religions it is a solar symbol.  An ancient Hungarian song about the "Wondrous Stag" includes the lines:

Amongst its horns it carries the light of the blessed sun
...
That it may be the messenger of heaven and bringer of news
About our creator and caring God

In a Persian legend, a prince out hunting encounters a wondrous stag with blue fur, a color associated with heaven.

In Finland, the stag was a favorite animal of the queen of the underworld and lured the hero to his doom.

In Siberia, the heavenly reindeer (the big dipper) steals the sun causing six months of darkness in the arctic.  So the legendary hunter searching for the stag is actually trying to bring back the sun.

In Christianity, the stag is an enemy of the snake, or serpent so is considered a symbol of Christ.  When deer cross a river, they do so in single file and help each other.  This is seen as symbolic of Christians helping each other to escape from a worldly life to a spiritual one.

The White Stag, popular in Celtic and other old European traditions, indicated that the otherworld was near (death), but could also mean the start of an important quest (death of your old life, start of a new one, like the death card in tarot).

So, in summary, the stag or deer symbolized the spiritual sun or Logos.