πόσος, 'how many?', is a pronoun that acts in the same way as an adjective. Therefore, it must decline for gender, case and number. What's going on in this sentence is that a second-person you (έχεις) is the subject and cats (γάτες) are the object. πόσος is acting like an adjective for cats. The noun for cats is feminine, we're talking about cats in the plural, plus they're the object of the sentence, so πόσος must appear in its feminine plural accusative form, which is πόσες.
All adjectives operate like this and it does take some getting used to, but it's pretty easy once you get the hang of it. I think the new tree will cover adjectives in the plural and in different cases, as the current course doesn't really touch on them much, which is why this sentence might be a bit tricky.
πόσα is the neuter plural accusative form of πόσος. So we'd use that if I were to ask something like 'how many houses (σπίτια) do you own?'
There's a declension table for πόσος here, hopefully that will clear things up a bit.
Well, I'm a native Greek speaker so I can't really take credit for my command of Greek, I mostly use this course like the English from Greek, to improve my English, and I try to help on the discussions whenever I can :) Your command of Greek is great though, I thought that you were a native speaker as well :)