The final "e" in "classe" is not necessarily silent, exactly. One can pronounce it as a very unstressed schwa. The male voice does this pretty consistently and clearly. However, in this exercise, it sounds unusual, as you pointed out, as though he is saying, "class-on". I have reported it and recommend that others do, as well.
As I understand it, c'est is used for nouns (and noun phrases). (Not exclusively.)
But, for more depth, see also:
Just for clarity, I will describe my view on this sentence and some aspects of its grammar.
The gender of the possessive determiner/adjective (for example, sa / son) usually is in agreement with the possessed noun. (There are some exceptions, for feminine nouns which begin with a vowel, when mon is used rather than ma.)
In this sentence, the gender of the possessive adjective sa is determined by the noun classe rather than the person being described.
The difference which pigslew has used is not on the possessive adjective, but on le meilleur / la meilleure, which in this sentence is linked to the person (c', a demonstrative pronoun, which I would equate to the subject of the English sentence) via the verb est. C' by itself is no indication of gender, but pigslew means if we read la meilleure, it should be she, and if we read le meilleur it should be he.
The de sa classe then further qualifies (restricts, clarifies, situates) la meilleure / le meilleur, but is not going to affect the gender.
In my opinion. (I hope this helps and does not contain imprecisions.)
I'm reasonably sure that is correct for adjectives, but not for nouns.
There's one very important exception to note: c'est should be used when using an adjective to make a general comment about (but not describe) a thing or situation. In this case, use the masculine singular form of the adjective.
- C'est normal ? — Is this normal?
- Non, c'est étrange. — No, this is strange.
(Really I was hoping for a link which confirmed this more precisely, but hopefully these are ... sufficient to cling to.)