Specifically, what is the meaning of affaire in this sentence?
Affaire generally corresponds to “business” in the sense of something that somebody should be concerned about (the kind of business that one minds).
C'est l'affaire de tous = “it's everybody's business”. The phrase is a bit more idiomatic in French, it might be translated as “this is a public concern”.
Occupe-toi de tes affaires = “mind your own business”
J'en fais mon affaire = “I'm making this my business”, “I'm taking care of this”
Affaire is broader than “business” in this meaning, it can more generally mean something that is happening and is a concern of some kind.
La mystérieuse Affaire de Styles = “The Mysterious Affair at Styles”
L'affaire Dreyfus = “the Dreyfus Affair” (and affaire would generally be used in this sense in French even where English would use “case”)
C'est une affaire d'argent, de cœur, d'État... = “it's a matter of money, of love, of State…”
The word affaire has many other meanings, including some other meanings which can be translated as “business” in the economic sense:
Les affaires marchent bien cette année = “business is booming this year”
Il a monté une affaire (rare, one would usually use entreprise or (colloquial) boîte) = “he started a business”, “he started a company”
Le monde des affaires = “the business world”