Learn French: Ce sur quoi ce scénario se base est un secret. - What this script is based on is a secret.

Because the verb is se baser sur (to be based on). When "what" is the object of a preposition like sur, you cannot use ce que. Instead, you use ce + preposition + quoi, giving us ce sur quoi.

Can I put the preposition at the end and say ce que ce scénario se base sur?

No, that is a direct translation of English grammar that doesn't work in French. In French, you can never end a clause with a preposition. The preposition must always come immediately before the pronoun it links to, which gives us a structure closer to "that on which".

Why is the verb se base reflexive here?

The verb se baser sur literally means "to base oneself on". French frequently uses reflexive verbs like this to express ideas that take a passive form in English, such as "is based on".

Would I use sur quoi if I were talking about a specific book the script is based on?

No. We use ce sur quoi because the thing the script is based on is an unidentified "what" (ce). If you were referring back to a specific noun, you would use a form of lequel. For example: le livre sur lequel ce scénario se base (the book on which this script is based).