Try reading this sentence, spoken by Michael Wood in the PBS production The Story of India: "It's easy to forget the great voyages of Columbus and Vasco de Gama were to find India."
Did you stumble when you got to were? That's because you were expecting voyages to be the direct object of forget (it's easy to forget the voyages). Instead, the object of forget is the phrase the voyages... were to find... . If Michael had made proper use of the little word that, my wife and I would not have needed to rewind the DVR in order to figure out what he said.
When the helpful little word that follows a verb, it alerts readers that the direct object of the verb will be a phrase, not a noun. Omitting that in such situations misleads readers, often forcing them to back up and reinterpret what they have read thus far.
You won't stumble this time: "It's easy to forget that the great voyages of Columbus and Vasco de Gama were to find India."
That, preceding the great voyages, prevents you from reading voyages as the direct object of forget.
You don't want your reader stumbling around, having to back up and re-read, do you?
In your worthy quest to omit needless words, don't omit that when it can escort your readers more smoothly through your sentence.
When in doubt, use that.
