Woman is the abstract representation of a phenomenon, not a reference to any specific individual female of the human species.
Ok, that sentence isn't going to enlighten anyone much, is it?
If one claims they believe in fairies, or unicorns, it means that they believe these creatures exist. In the case of unicorns or fairies, this is indeed something that necessitates belief because there has been, so far, little actual proof that they actually exist.
On the other hand, believing in (the existence of) women seems absurd — there is no belief involved, nobody will deny that they exist (unless you have grown up and spent your entire live surrounded by men, and “women” are as mythical as unicorns of course).
Apart from a believe in existence, the verb believe can also express a more abstract feeling: a trust in the power, possibilities, skills or capabilities of something or someone.
On the esoteric level, this is expressed as a belief in “(the power of) love”, a belief in “the good in people”.
More down to earth, one can express belief in another person. If I tell a friend that I believe in him, I do not mean I am convinced that he actually exists! I mean that I think he can do what he wants or needs to do. I believe he has the capability to succeed.
The line from the song is somewhere in between those two: the author expresses a confidence in the power and capabilities of women in and of womanhood. He doesn't express any feeling about any specific individual person, but he treats "woman" as a more abstract description of femininity or, if you like, a symbol for all women in the world.
This is a contrast with the use of man to indicate humanity, nowadays including man and women: if we talk about the world of man or man's creations, we (nowadays) assume man to symbolise humans.