Beginning in the fall of 1957 which element is necessary to complete the sentence?

The sample above is a phrase, particularly a participial phrase. Remember that participial phrase is consist of words with the -ing or -en form plus its modifiers. This kind of phrase always functions as an adjective. The element  that is necessary to complete the sentence is an independent clause.

The answer is:

It helps listeners keep track of which woman is speaking.

By hearing the voice of the different characters the listener is more likely to keep track of what each one is saying, and it´s easier to identify who is saying what and with what tone, this is very important when trying to figure out the intentions of a character and what he really means.