Learn Turkish: Eğer hastaysa, gelmez. - If he is sick, he will not come.

Hasta ends in a vowel, so we need the buffer letter -y- before adding the conditional suffix -sa. So it becomes hasta + y + sa (if he/she/it is sick).

Does Eğer mean "if"? Do I have to use it?

Yes, eğer means "if", but the conditional suffix -sa on hastaysa already does that job. You don't strictly need to use eğer, but it is frequently added at the beginning of a sentence to signal early on that a condition is coming.

Why does the translation say "will not come", but the Turkish uses the aorist tense gelmez?

In Turkish, the aorist tense (like gelmez for negative, gelir for positive) is very commonly used for predictions or the expected natural outcome of an "if" clause. While you could use the future tense gelmeyecek, gelmez sounds very natural here for a general expectation.

How do we know this means "if he is sick" and not "if I am sick"?

Neither hastaysa nor gelmez has a personal ending attached to it. When there are no personal suffix endings, the subject defaults to the third person singular (o), meaning he, she, or it. If it were about "I", it would be hastaysam, gelmem.