Hello.

Я думаю, русским мужчинам абсолютно всё равно, на каком масле им готовят обед (или что там еще, крепы?) их женщины.

In some internet grammar I read this:
Perhaps the most common of the Russian subordinating conjunctions is the ubiquitous что. This conjunction marks a verbal complement, i.e. a sentence that functions as a direct object of a verb. For example, if you have a word for what you wish to express as a direct object, you simply put that word in the accusative case: Максим слышал музыку 'Maxim heard the music'. But what if there is no word for what Maxim heard. In that case you have to describe what he heard in a sentence. However, sentences cannot reflect case and direct objects must have case in Russian. So, in Russian you have to put in a 'dummy' pronoun to reflect case and the most common one is что: Максим слышал, что Горький произвёл скандал в Нью Йорке 'Maxim heard (that) Gorky created a scandal in New York'. Notice that the corresponding 'that' in English is omissible; in Russian it is not.
I think I am missing something. Already I have been puzzled by the fact that in some events, one may use a relative pronoun with no antecedent, as in Я понимаю, что ты хочешь сказать. I understand what you mean. It was hard already for me to accept this fact, as no grammar I've read (including the highly-praised "Comprehensive russian grammar" by Terrence Wade) does mention it.
Now I see an ellipsis of the subordinating conjunction which some say is mandatory. The more I learn about russian, the more grammar books seem to be absolutly unreliable.

Could someone tell me when and how the pronoun/conjunction что can be omitted? Or provide some clues?

Thanks.