Learn Italian: Lui non mangia e sta peggio. - He isn't eating and he is worse.

The verb stare is used to talk about health and how someone is feeling, which is why we ask Come stai? (How are you?). Since peggio (worse) refers to his physical state or health, Italian uses sta rather than è (from essere, to be).

Why does the English translation say "isn't eating" when mangia is the simple present tense?

In Italian, the simple present tense covers both "he doesn't eat" (a regular habit) and "he isn't eating" (an action happening right now). You don't need a special "-ing" form here; non mangia covers both, and the context tells us which one makes the most sense.

How does peggio connect to other words we use to talk about health?

Peggio means "worse." It is the irregular comparative form of the word male (bad/badly). Just as you would say sta male (he is/feels bad), you use sta peggio to say he is or feels worse.

Do I have to include Lui at the start of the sentence?

No, you could just say Non mangia e sta peggio. In Italian, the subject pronoun is usually dropped because the verb endings already tell you who is doing the action. Including Lui here just adds a little emphasis or makes the subject completely clear from the very beginning.