Learn Danish: Kvinden klæder sig hurtigt på. - The woman gets dressed quickly.

In Danish, getting dressed is a reflexive action. The dictionary form is at klæde sig på (literally "to dress oneself on"). Because the subject is kvinden (the woman), we use the third-person reflexive pronoun sig.

Why is på at the end of the sentence instead of next to klæder?

At klæde på is a separable verb. In a main clause like this one, the verb (klæder) stays in the second position, but the little preposition or "particle" (på) is pushed to the very end of the sentence. The reflexive pronoun and any adverbs get sandwiched in the middle.

Why does hurtigt have a -t at the end?

The -t is added because the word is being used as an adverb to describe how she dresses (quickly), rather than describing a noun. You typically make an adverb in Danish by adding a -t to the adjective form (hurtig).

If the sentence was about "me" getting dressed, would it still use sig?

No, the reflexive pronoun must match the subject. For "I" (jeg), you use mig, so it becomes: Jeg klæder mig hurtigt på. You only use sig for third-person subjects like he, she, it, or they.