Can someone please explain to me why ønsker sometimes means want and sometimes means desire? Like "jeg ønsker et svar" means "I want an answer" but this sentence means "she desires a horse"? Is it when it is used with the reflexive pronoun it becomes desire? What is the nuance between them? Why can I not say "she wants a horse"?
The most literal translation of ønske is to wish. With the added reflexive, it means to wish to own or have something physical -- often for birthday or christmas. So in this sentence, we get the girl wishes (to have) a horse (for christmas/her birthday). You can indeed say she wants a horse (hun vil have en hest) but it sounds a bit rough, like a demand: she demands to have a horse.
Without the reflexive, it is a polite but insisting way of saying you wish to have something non-physical, as in jeg ønsker et svar senest i morgen meaning something like I want to have an answer tomorrow at the latest.
The line between these two is very thin indeed, and people often add the reflexive onto the second use. Danes love their reflexive verbs! In most cases, you can simply strip it off and the meaning is exactly the same.
A third meaning comes with forming the conditional, where you express that you wish something were true: Jeg ville ønske at jeg var højere (I wish I were taller, literally I would wish that I were higher).
PS: If any Danes come a cross this and have a better definition, feel free to correct me!
In English, though, it has come to be non-standard and even sounds comical ("I'm gonna run me a race" sounds like some sort of stereotypical hillbilly character in a cartoon). I think one reason why English speakers have such difficulties with reflexive verbs (and double negatives in the languages that have them), is that there is a stigma attached to those forms in our language.