Why do I enjoy strange movies so much?
Especially those in which the protagonist is a strange,
often socially awkward figure.

Because by all means I am a strange person myself, I always
feel somewhat uncomfortable and out of context with others,
and watching a movie that tells the story oftheir complex
yet full lifes gives me hope. Watching other people's
behaviour has also the double effect of both me entering 
the character and feeling a bit more "normal" as in "as
normal as the other people".

Until you really know them, but that's just the moral of
these movies. Everybody's strange so nobody actually is.
The classical fairy tale ending.