IMHO it has nothing to do with horse riding.
a riding hood is simply the hood that is attached to a sweater (i call the whole garment a hoody).
it is "riding" on your back if it is down. a hood proper would have been a seperate item, like a hat or a beanie.
probably because in ye olden times women were expected to wear hoods to cover their hair. and they wanted to distinguish the two.

@Martin Mayer: Etwas mehr Zusammenhang wäre gut zur Abklärung, bin mir aber ziemlich sicher, dass es "Rotkäppchen" ist.

Siehe auch:

http://www.fln.vcu.edu/grimm/redridinghood.html

@Della:

Ich glaube, Martin wollte keine Übersetzung, sondern lediglich wissen, *warum* Rotkäppchen so heißt, und was ein "riding hood" ist.

thank you very much, keewee! That's the answer to my question!
della, you were right, I just wanted to know WHY it had to be a "riding hood" in English!