Learn Hardcore Russian: На афише написано, что спектакль начнётся в семь часов. - The poster says that the play will start at seven o’clock.

Russian uses на because the information is understood as being written on a surface. A poster is treated like a flat thing with text on it, so на is the natural choice.

на афише = on the poster
в афише would usually sound wrong here

Also, after на in this meaning, афиша is in the prepositional case:

афиша → на афише

So the phrase is literally on the poster.

Why is it написано and not something like написан or написала?

Написано is being used in an impersonal construction meaning something like:

it is written
it says

There is no specific person doing the writing in the sentence, and the focus is on the written information itself. In Russian, this kind of idea is often expressed with a neuter-form predicate like написано.

So:

На афише написано, что... = On the poster it says that...

You do not need an explicit subject like it the way English usually does.