He's asking for examples of etiological texts....e.g., in the old testament, the story of Abraham and Isaac that explains the cultural practice of circumcision. From Greek mythology, there's the story of Apollo and Daphne, which accounts for the practice of awarding laurel wreaths, among many others. From Greek history, the battle at Marathon against the Persians explains the race that carries the name.

I think your professor is asking you to describe how those myths reflected the life/society of the author, in those subjects. For instance, I'll use Marvel superheroes. Cultural institution could be how Ironman can represent the military industrial complex, often wages war with foreign powers, and is absolutely decadent in his personal life. Tony Stark is everything America shouldn't be. Another example could be how Matt Murdock is lawyer, who ought to uphold the letter of the law, but is also the vigilante Daredevil. This could reflect the distrust we have in the legal system.

I just made this stuff up off the top of my head, the comics, reality, and what they reflect are actually rather more involved and nuanced, but you're in an undergrad intro class. Something along these lines ought to work, only with ancient myths and what you know of ancient society

It looks like you are supposed to write about how the mythic past and the legendary/historical past explains some of the authors of your sources (Herodotus and Livy for example) beliefs and practices. So how does what Herodotus wrote influence his own beliefs.

At least that is how I interpreted the question.