Quite grammatically correct! But it would be used very rarely, at least in the US. It would pretty much universally be, "What are you listening to?"
Still -- the grammatically correct version should at least be accepted as an alternative. Some people do try to keep infinitives together even in speaking, not just in writing.
If this was a course in common English usage, I'd agree. I don't see a reason not to include it as an acceptable alternative for translation from Czech though.
Even the Cambridge Dictionary blesses the use of "to" at the end of the sentence. The following sentence is provided as the first example under its first definition of "listen": What kind of music do you listen to? Source: https://dictionary.cambridge.org/dictionary/english/listen. As noted elsewhere, the course attempts to strike a balance between what may be (or at one time was) technically grammatically correct but is rarely used, and what is actually used in the real world.
Does "poslouchat" mean "listen" or "listen to"? In another exercise, it was "listen" but here it seems to be "listen to." Can it mean both?
It can mean both. English requires "to" for the verb "listen" most of the time, but Czech "poslouchát" does not require a particle, so it can replace both. Poslouchám = I am listening. Poslouchám rádio = I am listening to the radio.
From the English side, that would just be wrong "What are you listening to" is the most common usage, but it's not quite right grammatically. "To what are you listening" is grammatically correct, but not much used (so it sounds... odd.).
You might also hear "What do you listen to," but I think that would more often be used to ask a general question about what someone LIKES to listen to or OFTEN listens to, rather than about what someone is NOW listening to.