Learn Farsi: گربه‌ی سیاه روی مبل است.

Words for locations like ru (on/top) act a bit like nouns in Farsi. To say "on the sofa", you are literally saying "the top of the sofa". This means you need to connect ru to mobl using the Ezafe. Because ru ends in a vowel, the Ezafe is pronounced as -ye, making it ru-ye.

Why does gorbe (cat) also have a -ye attached to it in gorbe-ye siyâh?

This is another Ezafe! Here, it connects the noun gorbe to its adjective siyâh (black). Just like ru, the word gorbe ends in a vowel sound (a short 'e' sound), so the Ezafe linking them together is pronounced and written with a -ye.

The English translation says "The black cat", but I don't see a word for "the" here. How do we know it is specific?

Farsi does not have a direct word for "the". A noun phrase like gorbe-ye siyâh can mean "a black cat" or "the black cat" depending on the context. Since it is the main subject of this sentence and we are talking about its specific location, it is understood as "the" black cat.

Why is the verb ast (is) placed at the very end of the sentence?

Farsi follows a Subject-Object-Verb word order, meaning the action or "to be" verb almost always goes at the very end of the sentence. The sentence flows as: subject (gorbe-ye siyâh), location (), and finally the verb ().