Learn Hardcore Russian: Кассир попросил меня сохранить чек. - The cashier asked me to keep the receipt.

Because the subject кассир (cashier) is grammatically masculine, so the past-tense verb agrees with it: кассир попросил.
If the cashier is a woman, you can still say кассир попросил (profession nouns are often masculine by default), but it’s also common to use feminine agreement in real speech: кассир попросила.

What case is кассир in, and why?

Кассир is in the nominative case because it’s the subject (the one doing the asking). It answers “who?” in the sense of “who asked?”

Why is it меня and not я?

Because попросить takes a direct object: “to ask someone.”
So you need the accusative form of я → меня (“me”).
Pattern: попросить кого? + infinitive / or + о чём (about what).

Does попросить always take the accusative like this?