Learn Spanish (Spain): Estoy enferma y triste.

Both verbs mean "I am", but we use estar (like estoy here) to talk about feelings, health, and temporary states. Since being sick and being sad are physical and emotional conditions, estoy is the correct choice.

Why is it enferma instead of enfermo?

In Spanish, adjectives must match the gender of the person they describe. Because the speaker says enferma (ending in -a), we know the person speaking is female. A male speaker would say Estoy enfermo.

Since the speaker is female, why doesn't triste change to end in -a too?

Adjectives that end in -e, like triste (sad) or inteligente (smart), do not change for masculine or feminine. They are exactly the same for men and women. They only change when making them plural (like tristes).

Where is the word for "I" (yo) in this sentence?

In Spanish, you can usually leave out the subject pronoun because the verb itself tells you who is doing the action. The verb form estoy only ever means "I am", so adding yo at the beginning is optional and often dropped to sound more natural.