The way i understand this without using all that grammatical jargon, is that when the sentence at any other point reveals the person's (or groups) gender such as it did in this one with "fille" and then "elle" then you use "C'est" (or "Ce sont"). If you are talking about anything else that involves a person or people then you use "Il, Elle, Ils, Elles" as in "Elle est calme"! (no secondary gender reveal)
As it states at the beginning of the lesson: The basic rule is that you must use ce when être is followed by any determiner—for instance, an article or a possessive adjective. Note that c'est should be used for singulars and ce sont should be used for plurals.
C'est un homme. — He's a man. / This is a man. / That is a man. Ce sont des chats. — They're cats. / These are cats. / Those are cats. C'est la fille. — She is the girl. / This is the girl. / That is the girl. Ce sont les femmes. — They are the women. / These are the women. / Those are the women. If an adjective, adverb, or both appear after être, then use the personal pronoun.
Elle est belle. — She is beautiful. (Or "It is beautiful.") Il est très fort. — He is very strong. (Or "It is very strong.")
Why does she pronounce it like 'cette' (set) instead of c'est (say)???
That's correct actually. It's like "the" (tha) vs "the" (thee). If the first pronounced letter of the next word is a vowel, its pronounced "thee", if the first pronounced letter is a consonant, it's "tha".
So in French, if the first pronounced letter on the next word is a vowel, "c'est" is pronounced like "set". If its a consonant it's pronounced like "say".
Because its sort of inferred. I think its more of a politeness thing. You wouldn't say "It is a girl." That is kind of an insult in English. So you infer from the rest of the sentence that "she" is feminine and can say "She is a girl". Most things is languages do not perfectly translate, so try not to set it too rigidly on "C'est = it is"