Learn Hardcore German: Der Schüler schreibt dieses Zitat in seinen Aufsatz, um eine bessere Note zu bekommen. - The pupil writes this quote in his essay in order to get a better grade.

Because Der Schüler is the subject of the sentence. In German, the subject always takes the nominative case, and der is the masculine nominative singular article.

Why is dieses Zitat in the accusative case?

dieses Zitat is the direct object of the verb schreibt, so it must be in the accusative. The neuter noun Zitat takes dieses in the accusative singular, matching its gender and number.

Why does seinen Aufsatz use seinen instead of sein?

Aufsatz is a masculine noun, and here it’s also a direct object (accusative). The possessive pronoun sein (his) must take the strong accusative ending -en for masculine nouns, giving seinen Aufsatz.

What is the function of the um … zu construction?

The um … zu infinitive clause expresses purpose. It literally means “in order to.” So um eine bessere Note zu bekommen translates as “in order to get a better grade.”

Why is there a comma before um?

In German, you separate the main clause from an infinitive purpose clause introduced by um with a comma. Every um … zu clause must be preceded by a comma.

Why does bessere end with -e in eine bessere Note?

Note is feminine, and eine is an indefinite article. In the accusative singular for feminine nouns, adjectives take the -e ending, so eine bessere Note is correct.

Why is eine bessere Note in the accusative case?

Because bekommen requires a direct object—the thing that is “gotten.” That direct object is eine bessere Note, so it appears in the accusative.

Can I use damit instead of um … zu bekommen?

Yes. damit also expresses purpose but introduces a finite subordinate clause. You’d say:
Der Schüler schreibt dieses Zitat in seinen Aufsatz, damit er eine bessere Note bekommt.
Note that with damit, you need a subject (er) and a conjugated verb (bekommt) at the end of the clause.