Well, in a way it's innate for us guys to protect women and want to rescue them. So naturally there's a part of us who would want to 'be there" for a girl who is needy. The danger there is, it's not a healthy relationship and often causes more problems than you started with.
As for the women not needing anyone, if you want to simplify it you could say it like this. Looking for a relationship is a person's kind of saying "I need someone else. I want that special someone else in my life because I feel that need". That is not to say a man or women can truly fulfil that longing, because only God can. But it's still a basic need. By saying that they 'dont need a man" or giving someone the impression that they are better off by themselves, what encourages the man to try and form a relationship with that person? A relationship is both of you bringing to it things you have learned, things that are a part of you, your personality, your traits, to compliment each other and work together. When a women through her actions or words says "I dont need you" then why would a man be interested in going into a relationship where he feels that he is not bringing anything to it?
And I would argue that the woman (and I'm not just trying to pick on women, this is just the example you were giving) who acts or speaks in such a way that she doesn't need others, is again, lying to herself. She is afraid to feel vulnerable and that she does need someone else.
Guys do the same thing. Just in different ways.