Verb patterns with that-clauses

That is simply a connector. That-clauses can have various functions in sentences. A that-clause, for example, can be the subject or object of a verb.

Verb pattern: subject + verb + that-clause

We hoped that we would win.
The teacher said that honesty is the best policy.
Mother suggested that I should consult a doctor.
He admitted that he had stolen the watch.
They complained that they had not been treated fairly.
We knew that the next day would be difficult.
I regretted that I was not going to be at the meeting.
I knew that she was innocent.

Common verbs that can be followed by that-clauses are: say, think, imagine, suppose, know, believe, admit, confess, suggest, complain, hope, expect, fear, feel, hear, show, understand and wonder.

Note that that is often omitted especially after verbs like say, think, suppose, hope and expect.

Alice said that she was feeling better. OR Alice said she was feeling better.
I thought that I would be late. OR I thought I would be late.

In an informal style that is sometimes left out in some common two-word conjunctions such as so that, such…that, supposing that etc.

Hurry up so (that) we won’t miss the train.
You can borrow my car provided (that) you bring it back tomorrow.

A that-clause can also follow a noun / pronoun.

Subject + verb + noun/ pronoun + that-clause

He told me that he was coming.
I warned him that he shouldn’t repeat it.
She assured me that she would help.
They have informed us that they are leaving this afternoon.
He convinced me that he could do the work well.