Learn Spanish (Latin America): ¿Estás cansada? Sí, lo estoy.

While lo often means "him" or "it" as a direct object, here it acts as a neuter pronoun replacing the adjective cansada. In English, we can end a short answer with "I am," but in Spanish, verbs like ser and estar cannot be left hanging without their attribute. The lo stands in for the omitted adjective.

Since the speaker is female (indicated by cansada), shouldn't the pronoun be la instead of lo?

No, when you use a pronoun to replace an adjective, you must always use the neuter lo. It never changes to match the gender or number of the subject. Even if a group of women were asked if they were tired, they would still answer, "Sí, lo estamos."

Can I just say "Sí, estoy" and drop the lo?

No, saying "Sí, estoy" sounds incomplete and unnatural in Spanish. If you do not use the pronoun lo to represent the state, you have to repeat the adjective itself by saying, "Sí, estoy cansada."

Do native speakers actually say "Sí, lo estoy" in casual conversation?

They do, though it can sound slightly emphatic or formal depending on the context. In everyday casual speech, it is very common to simply reply with a quick "Sí" or to just repeat the adjective.