This is a good question. The answer is straightforward, if not entirely simple.
This sentence translates literally to "The house is there, where the cat is," but that doesn't make sense in English. In English we don't need the adverb (there) before the relative pronoun (where), so we eliminate the "there." Ukrainian requires it. If we took it out of the original, we would end up with Дім де кіт, which would mean something like "House where cat." Because Ukrainian doesn't use copulas (for our purposes, "be" verbs), this sentence doesn't make sense without that там to guide us.
In English, structures like this are called free relative clauses and they are quite ordinary: I like what you did; She ate what she wanted; We can hear what you're saying. These sentences can't exist in Ukrainian without antecedents. For example, Мені подобається те, що ти зробила (I like that, which you did = I like what you did); Вона їла те, що хотіла (She ate that, which [she] wanted = She ate what she wanted); Ми чуємо те, що ви говорите (We hear that, which you are saying = We [can] hear what you're saying).
When I read the duolingo sentence my mind wants to think exactly what you said. "The home is here, where is the cat?" When does this sentence come into play in Ukrainian speech? Maybe they could say it isn't home without the cat? I didn't know that they place the bed where the cat lays.