"My house is between these houses."
Translation:Mein Haus ist zwischen diesen Häusern.
First off, obey the capitalization rules—Häuser, not häuser.
Second, to answer your question: Because zwischen is used, you have to make what your house is between (these houses) into the dative case. The dative plural of Haus is Häusern (and dative plural of dieser is diesen). Hence, zwischen diesen Häusern.
I agree that it's kind of crappy Duolingo requires you to know this to get your questions right before even introducing the concept of inflecting nouns in anything besides plurality.
"Zwischen" is one of several "dual" prepositions that take either the dative or the accusative depending on meaning. The dative version is static - whatever action is going on between the nouns in question, it's there for the whole sentence. The accusative version is more dynamic - the action starts out not between the nouns, and then moves between them. So if you're standing between two houses, you would say "Ich stehe zwischen den Häusern." But if you're trying to avoid being seen and suddenly duck between two houses, you say "Ich ducke zwischen die Häuser". (Whereas "Ich ducke zwischen den Häusern" would mean you started out standing between the houses and then suddenly decided to duck..)
It can, but it changes the meaning. If you are standing between two houses, then you use dative. If you quickly duck between two houses, then you use accusative. The difference between dative and accusative in these cases is always motion/change: if the preposition applies throughout the action described by the sentence, use dative.. but if the action includes a change that causes the preposition to start applying (you weren't between the houses, but now you are), then use the accusative. This "accusative of motion" is a recurring bit of grammar across many languages, including Latin.
This is called the "N-Deklination" in German grammar. It applies only to masculine nouns, and then only to those "weak" nouns that change their ending depending on the case, e.g. "der Mensch" in the four cases: der Mensch, den Menschen, dem Menschen, des Menschen. I don't fully understand the logic behind it, but there's a good overview here:
It seems that the speaker is looking at three houses, and saying his is the middle house. But, it's probably just a sentence to see if we give diese the correct ending, or testing our understanding of zwischen, or the spelling in German for the plural of "House". I depends on what section this lesson was in. ;-)