Q:

In what cases can 'how' replace that?

My mom and I both got the iPhone 7+ yesterday. She just text my sister
  and I how she doesn't know how to screenshot on this phone
He told us how he was honest and could be trusted

A:

I guess it depends on the situation, and if people actually use the term correctly for what they are trying to say.

He told us that he was honest and could be trusted

He doesn't give you an explaination to why he is honest - just that he is.

He told us how he was honest and could be trusted

He tells you how he is honest by gives you examples to convince you, that he is, in fact, honest.
I am no linguistic wizard - but this is how I interpret it, or at least how it should be used.

