Learn Hardcore Russian: У неё болит горло.

Russian usually does not say “She has …” with a verb like to have.
Instead it uses a structure «у + person (in genitive case)» + [something] that literally means “at/near/by someone there is [something]”.

У неё болит горло. = At her the throat hurts → Her throat hurts / She has a sore throat.

So «у неё» here is the normal Russian way to express “she has …”, including “she has a pain / she has something hurting”.

What case is «неё» in, and why that case?

«неё» is the genitive singular of она (she).

The preposition «у» always takes the genitive:

у меня (of me)
у тебя (of you, informal)
у него (of him)
у неё (of her)
у нас (of us), etc.

So «у неё» is “by/at her” in a grammatical sense, but functionally it means “she has …” in this type of sentence.

What does «у» literally mean here?