"They hear a horse."
Translation:One słyszą konia.
First of all, 'koń' is the direct object of this sentence, so you need (usually) the accusative case.
For masculine animate (i.e. describing living things) nouns, the accusative is the same as genitive. Genitive for masculine animate and neuter nouns ends with '-a', hence 'konia' (ń becomes ni because you can't have ń in front of a vowel).
What in the world are masculine animate, accusative, genitive, and neiter nouns?
In Polish there are three types of nouns which are called "gender": masculine, feminine and neuter. "Animate" means just "living" so masculine animate nouns name people and animals. Accusative and Genitive are cases. You can read more about them for example here: http://mowicpopolsku.com/polish-grammar/cases/ or here: https://en.wikibooks.org/wiki/Polish/Noun_cases
Here you can find a simplified chart of declension: https://i.imgur.com/NjFhVJV.jpg
Why doesn't Duolingo teach us this? I feel like this is never something I'll understand without additional research on my own.
It does teach us, but only on the website. There there are grammar files that explain this quite well.
You "listen" to something/someone actively, but you just "hear" because your ear catches the sound.