Learn Hardcore Hungarian: A pénztáros azt mondja, hogy ma akció van, ezért a sajt kevesebbe kerül. - The cashier says that there is a sale today, so the cheese costs less.

A is the definite article, meaning the. Here the speaker is talking about a specific cashier in the situation, so Hungarian naturally says A pénztáros = the cashier.

Using bare pénztáros here would not sound like a normal complete subject in this sentence.

Why is the verb mondja and not mond?

Hungarian has two main verb conjugations: indefinite and definite.

mond = he/she says in the indefinite conjugation
mondja = he/she says it / says that in the definite conjugation

Here, what is being said is treated as a definite object, shown by azt and the following hogy clause, so mondja is the natural form.

What is azt doing in azt mondja?

Azt literally means that or it in the accusative form. In this sentence, it points forward to the whole clause after it:

azt mondja, hogy... = says that...

This is very common in Hungarian. English usually just says says that..., but Hungarian often uses this extra pointer word.