I would speculate that the fact that we sound more high-pitched than what we think we should leads us to cringe as it doesn't meet our internal expectations;(original)
Question: Is the original grammatically correct?
I can not find the reason by grammar to explain why the "what" is used?
What does the pronoun "it" here refer to?

Question

I would speculate that the fact that we sound more high-pitched than what we think we should leads us to cringe as it doesn't meet our internal expectations;(original)
Question:  Is the original grammatically correct?
I can not find the reason by grammar to explain why the "what" is used?
What does the pronoun "it" here refer to?

Answer

Yes, it is grammatically correct.

"what"= direct object of verb "should"
"what" replaces "sound like."

Original:
... than what we think we should ...
Equivalent:
... than we think we should sound like ...

"it" = subject of verb "doesn't meet"
"it" replaces "the fact that we sound more high-pitched."

Original:
.... as it doesn't meet our internal expectations.
Equivalent:
... as the fact that we sound more high-pitched doesn't meet our internal expectations.
or ... as sounding high-pitched doesn't meet our internal expectations.

Answer

Why don't,t you try it and I'll correct it.

Answer

Not technically speaking grammatically correct, no. 'What' should be omitted. The sentence should be clarified, and cut into two shorter parts. 
'It' refers to the sound of 'our' speaking.