Translation:There is neither a cat nor a horse here.
Ни uses the genitive case. Кошки and лошади are indeed plural forms, but they're in the nominative case. The plural nominative and singular genitive forms of these two words just happen to be the same. The plural genitive forms would have different forms, hence why it's singular. :)
"Here" is used as a locator, to define where something is, not as the noun/subject of a sentence. It may seem overly particular, but "there" is the correct indefinite subject/noun to use, while still using "here", as in "There is neither a cat nor a horse here." (American) English simply doesn't use "here" in the way that it uses "there".
It's not easy to define when a silly mistake becomes an incorrect translation, nor is it easy to program a system that can account for all possible minor errors. It's unfortunate that English is the base for many of the languages taught here, especially since English is not any easy language in many regards. But for now it's the international language of choice, mostly for historical reasons based on the British Empire and American international influence arising from World War II and the ensuing cold war with the Soviet Union. If WW II hadn't happened, I doubt that English would be so wide-spread.
I'm a native-English speaker, but I rely heavily on spell-check to catch my frequent spelling mistakes, and I usually go over most of my posts to correct my grammar. I don't know whether it's possible, but you might consider installing an English-grammar checker on your computer. That might be much more trouble than it's worth, but it's an idea.