Learn Russian: Ритм этой музыки оказывает сильное влияние на слушателя. - The rhythm of this music exerts a strong influence on the listener.

Both mean to influence, but оказывать влияние (literally to render or exert an influence) is a formal collocation used in written or journalistic Russian. At the B2 level, you will often see the verb оказывать paired with abstract nouns, such as оказывать поддержку for providing support or оказывать давление for exerting pressure.

Which case is used for слушателя after the preposition на?

It is in the Accusative case. Because слушатель (listener) is an animate masculine noun, its Accusative form mirrors its Genitive form, taking the -я ending.

Why does the phrase влияние на take the Accusative case here instead of the Prepositional case?

Even though на takes the Prepositional case to show static location, it takes the Accusative case when showing the direction or target of an action. Here, the influence is being conceptually directed onto the listener.

How would the sentence change if the music influenced multiple listeners?

You would need the plural animate Accusative form. Because animate Accusative matches the Genitive for plural nouns, слушатели changes to слушателей, making the phrase на слушателей.