Audi is a man's surname, translated into Latin because he was not allowed to use it.
August Horch built cars under his own name. Zwischengas, a German car history site, reports that in 1909 he left the Horch company he had founded after disputes with its supervisory board. Soon afterward he registered a new firm in Zwickau as August Horch Automobilwerke GmbH.
The old Horch company objected to a second car brand with the same name and sued. Horch lost. Wikipedia describes it as a trademark dispute that barred him from using "horch" as a trade name for his new firm.
He needed a new name that still meant the same thing. In German, Horch is the imperative of the verb "to listen." It means "hark" or "hear." The Latin verb for the same idea is audire, and its imperative is audi. Etymonline's entry for the prefix audio- derives it from Latin audire, "to hear."
The story of who thought of it is charming. At a meeting with friends Paul and Franz Fikentscher, Franz's son, who was studying Latin, offered the translation. Zwischengas gives his words as: "Father, audi et altera pars, wouldn't it be right to say Audi instead of Horch?" Both sources say the idea came from the boy, though the quotation rests on a retelling, so it should be read as the family legend of the firm.
The company was entered in the Zwickau register on April 25, 1910, as Audi Automobilwerke GmbH.
The four rings came later. Wikipedia says they stand for the four companies that merged in 1932 to form Auto Union: Audi, Horch, DKW and Wanderer. So the Horch name came back, but on a badge shared with three other firms.
The result is an unusual case in branding: a real surname, not an invented word, carried into a new language.