Learn French: Le directeur annonce qu'il part demain. - The director announces that he is leaving tomorrow.

No, you can't. In English, we often drop the word "that" and simply say "The director announces he is leaving." In French, the conjunction que (that) is absolutely required to link the main phrase with the second part of the sentence. You can never skip it.

Why is it written as qu'il instead of que il?

In French, when que is immediately followed by a word starting with a vowel or a silent h, the e is dropped and replaced with an apostrophe. Since il starts with the vowel i, que il must become qu'il. This is called elision, and it makes the phrase flow much more smoothly when spoken.

Why does the sentence use the present tense part if he is leaving demain (tomorrow)?

Just like in English when we say "he is leaving tomorrow," French often uses the present tense to talk about a planned, definite event in the near future. This happens almost always when you include a clear time word like demain. You could also use the near future tense and say qu'il va partir demain, but using the simple present tense is very common and natural here.

Could I use the verb sort or quitte instead of part for "leaving"?