"How is it that she shouted" is a bit of a strange sentence. "how is it that" is an expression, and it basically asks how something works or how something is possible.
- How is it that everyone loves Taylor Swift?
- How is it that professional tennis players can play for hours even in the heat?
So your example sentence sounds like it's incomplete, like you wanted to say something like
How is it that she shouted when she was barely even whispering?
That's an example of someone saying "your friend shouted" when your friend was actually speaking quietly, so in effect you're asking "when someone is speaking quietly, how is that an example of shouting?"
Otherwise, just "how is it that she shouted" implies that she shouldn't have been able to shout (for example, she underwent throat surgery), so you wonder how she was able to shout anyway. If that's your intended meaning, sorry for the above long explanation :)
Since "how is it that" is an expression, you can't just drop the "it".