Learn Russian: Соседи заговорились о политике на кухне. - The neighbors got carried away talking about politics in the kitchen.

This combination creates an "absorption" or "excess" verb. When you attach both за- and -ся to a verb, it expresses getting completely absorbed in an action to the point of losing track of time. Thus, заговориться means to get carried away chatting.

Since they were talking for a long time, why is заговорились a perfective verb?

Even though the action of talking was continuous, the verb focuses on the result: the moment they crossed the line into being "carried away" or "absorbed." That shift into a state of excess is viewed as a single, completed event, making it perfective.

Could I use разговорились instead of заговорились in this sentence?

You could, but it changes the meaning slightly. Соседи разговорились focuses on the conversation gaining momentum (they got into a good, flowing conversation). Соседи заговорились focuses on excess (they talked for too long and lost track of time).

Why is it на кухне instead of в кухне for "in the kitchen"?

While в кухне (literally inside the room) is grammatically acceptable, на кухне is the traditional and overwhelmingly common way to say "in the kitchen" in everyday Russian. It historically treats the kitchen as a functional workspace or zone rather than a closed box.