Learn Russian: Что ни кричит публика, комик не уходит со сцены. - Whatever the audience shouts, the comedian does not leave the stage.

This sentence uses a related concessive clause (что ни - whatever) to express the exact same "despite" idea. You could easily rewrite the first half using the lesson's target grammar by adding то, что to bridge the preposition and the verb: Несмотря на то, что кричит публика, комик не уходит... (Despite what the audience is shouting...).

Does the ни in что ни кричит make the action negative?

Not at all! When you combine a question word (like кто, что, где) with the particle ни, it creates the meaning of "no matter [who/what/where]" or the English suffix "-ever". So, что ни кричит simply means "whatever they shout" or "no matter what they shout".

Could I use не instead of ни here?

No, because it would completely change the meaning. If you wrote что не кричит публика, it becomes a simple negative relative clause: "what the audience is not shouting". The particle ни is specifically required to trigger the "whatever" meaning.

Why is it со сцены instead of из сцены?