"The woman is listening to the radio."
Translation:Kobieta słucha radia.
I don't know if there's a better answer than "it just does"... to me it is obvious, and if for a minute I wasn't sure, I ask 'the questions' for cases: słuchać kogo? słuchać czego? makes sense; while słuchać kogo? słuchać co? does not. But that's rather for Polish people, of course. Because we do get confused about cases as well sometimes.
"talking to" - none of those, you need a preposition. "Mówię do kobiety" - and 'do' always takes Genitive.
Similarly "walking to" = Idę do domu (I'm walking/going home / I'm walking/going to a/the house).
Different verbs take different cases. The biggest part of them take Accusative, and in the beginning of the course, you learn almost only such verbs. I understand that you now started encountering verbs taking Genitive. And well, it's hard to say 'why' the verb takes Genitive, it just does. Just like some prepositions take it. Sometimes you may 'rewrite' the verb in a way like: "to need" = "to have the need OF" - and 'of' takes Genitive, so that's some help. But not with "listening to". And unlike English, the Polish equivalent doesn't use any preposition.