Your sentence becomes more clear when you use "that". You can omit it, but in a sense it remains there whether or not you use it. The sentence will be understood as if you had said "that", whether you say it or not. Your sentence shows the use of the use of the conjunction "that" to introduce subordinate noun phrases. "That" can turn ANY sentence into a noun! For example: "You are wrong" is a sentence but "that you are wrong" is a noun phrase. "The way to know is to ask" is a sentence but "that the way to know is to ask" is a noun phrase.

The word "that" has different definitions. It looks like this is the word "that" as a conjunction. What is a conjunction? A conjunction is a word that links other words, phrases, or clauses together. Hope this helps.

As a general rule, these verbs followed by an assertion/proposition/verbal phrase can always be followed by 'that', and native speakers omit 'that' because it is easier/quicker to say. It is very common to omit 'that' with frequent verbs (know, say, think, etc.) and less common with more formal/ longer words (inform, convince). If in doubt, I would keep 'that' because that will always be correct.

in such sentences it's up to you -- can use or can skip

Sad and sad