If there is no danger of misreading, the word that may be omitted when it introduces a subordinate clause. The value of a principle is the number of things [that] it will explain. When a sentence might be misread without that, however, it is necessary to include the word.

Milgram didn’t discover ordinary people; he discovered that ordinary people were willing to inflict pain on strangers. The word that tells readers to expect a clause, not just ordinary people, as the direct object of discovered.