Before having anything to do with geeks and fans, "otaku" was an archaic form of very polite address literally meaning "your house".
Then some fantasy geeks started addressing each other, and occasionally other people by that term. And people started calling the kind of people who did that otaku-jin, or just otaku.
So imagine some ren-faire geek who talks to friends, or even random people, in pseudo-Shakespearean english. That's the americanized equivalent of the kind of person who would be considered otaku. They're extremely geeky, but not necessarily hikikomori types.
The word may have acquired some different connotations since then, both in japan and america, but that's how it started.