That is simply connector. It shows that a clause forms part of a larger
sentence.

I know. He is innocent. (Two separate sentences)
I know that he is innocent. (One larger sentence in which the clause ‘he is
innocent’ becomes the object of the verb know.)

A that-clause can be the subject or object of a verb.

That she should forget me so quickly was rather a shock. (Here the
that-clause ‘that she should forget me so quickly’ acts as the subject of
the verb was.)

I knew that he wouldn’t come. (Here the that- clause ‘that he wouldn’t come is the object of the verb knew.)

Preparatory it

It is often used as a preparatory subject or object for a that-clause.

It surprised me that he was not yet ready. (More natural than ‘That he was not yet ready surprised me.)

Verbs in that-clauses
After certain words like insist, demand, suggest, recommend, ask, vital, important and essential, should + infinitive is used instead of ordinary verb forms.

Mother suggested that I should consult a doctor.
She insisted that I should go at once.
It is important that the meeting should start on time.

Omission of that
In an informal style that is often left out.

She said she didn’t understand. OR She said that she didn’t understand.
It is strange he hasn’t written. It is strange that he hasn’t written.
I thought you were in New York. I thought that you were in New York.