Learn Russian: Соседи снова зашумели в своей квартире. - The neighbors started making noise in their apartment again.

The prefix за- added to the verb шуметь (to make noise) creates the perfective verb зашуметь. In Russian, adding за- to certain verbs—especially those about sounds, emotions, or weather—means the action suddenly or spontaneously started. So, зашумели literally contains the meaning "started making noise."

Could I just say Соседи снова начали шуметь?

Yes, you could! Начали шуметь is grammatically correct and means exactly that. However, using the prefixed verb зашумели sounds much more natural and native. It emphasizes that the noise started abruptly, which perfectly fits the context of annoying neighbors.

Since шуметь is an imperfective continuous action, what is the aspect of зашумели?

Зашумели is perfective. Even though the noise itself continues, the start of the noise is viewed as a single, completed event in Russian. Once they have crossed the threshold into being noisy, the action of "starting" is complete.

Why does the sentence use в своей квартире instead of в их квартире?

The reflexive possessive pronoun свой (one's own) is required here because the owners of the apartment are also the subject of the sentence (соседи). If you said в их квартире, it would grammatically imply the neighbors were making noise in someone else's apartment!