Good discussion here...thanks for bringing up this issue. The first thing to explain here is what a "reporting verb" is. Essentially, it's a verb that reports or otherwise includes a thought or belief. Examples of "reporting verbs" are indicate, claim, announce, or report.
When you are reporting a clause (includes a verb), you need "that":
"The students reported that they felt great."
"The lab results indicate that your theory is correct."
"The King claimed that he had turned lead into gold."
When you don't have a verb following the word, you don't need "that":
"I reported my friend for cheating."
"I indicated disbelief."
"The defendant claimed innocence despite overwhelming evidence to the contrary."
Does this help clear up the issue?