Not as much as one would initially think! Shakespeare's particular rendition (there were others!) digresses from the stereotype role fairies were cast into (naughty little imps bent on meddling in human affairs) by making them more a 'reflection' or 'shadow' of the psyche - in other words, the an expression of the subconscious:
If we shadows have offended,
Think but this, and all is mended.
That you have but slumb'red here
While these visions did appear.
And this weak and idle theme,
No more yielding but a dream
(V, i, ll. 412-417).
The ambivalence here of the fairies' role (insubstantial and orinique) and even the reality thereof is altogether intentional; it's as if the reader can choose which 'parallel world' he wants to believe in (in terms of the story line, that is). There is a wide margin for interpretation, and this is the primary charm of the story.
The following references will give you further insight into this "magical" tale.
Please see the link below for another answer.