Learn Russian: Она кажется усталой.

The verb казаться (to seem) is a reflexive verb, which you can tell by the -ся ending. In this case, it doesn't mean she is doing an action to herself. Instead, Russian often uses reflexive endings for verbs that describe a state, appearance, or an impression.

Why is the adjective усталой in the Instrumental case?

Verbs that describe seeming, being, or becoming (like казаться) require the adjective or noun that follows them to be in the Instrumental case. Because она is feminine, the adjective changes from the Nominative усталая to the Instrumental усталой.

Could I just use the Nominative case and say Она кажется усталая?

No, that would sound incorrect to a native speaker. While you use the Nominative for a direct statement like Она усталая (She is tired), adding the verb казаться acts as a grammar trigger that forces the descriptive adjective into the Instrumental case.

Why isn't the reflexive pronoun свой used in this sentence?

The pronoun свой is used to show possession (meaning "one's own") when the owner is the subject of the sentence. Since this sentence only describes her physical state (tired) and doesn't mention any objects, family members, or traits that she owns, there is no place for a possessive pronoun here.